One-dimensional Granular Chain Research Articles

Dynamic behavior of one-dimensional foam disk chains during low-velocity impact

This study experimentally analyses the deformation and energy retention behavior of ten closed-cell PVC foam disks constituting a one-dimensional granular chain subjected to low-velocity impacts. A pressure-driven impactor loads disk chains of two densities, each at 1 and 10 m/s. The deformations of the disks are captured using high-speed cameras, and the chain's input and output forces are recorded during the impact. Energy retained by the chain and the disks is calculated using the deformations of the disks, input, output, and contact forces. The results show that the energy retained by the chain of H130 foam is 3 times greater than that of the H35 foam at 1 m/s and is about twice as high at 10 m/s. Normalized with respect to input energy, the relative energy retained by the chains of H35 foam is more than that of the H130 foam for the same impact velocity and is higher at lower impact velocities for both foams. In addition, the energy the disk chain retains decays along the chain, and the decay rate is higher for lower impact speeds, implying that the disk chain should be longer for higher impact speeds to mitigate the energy fully.

One-dimensional granular chains as transmitted force attenuators

One-dimensional granular chains as transmitted force attenuators

Detection of defects in cellular solids using highly nonlinear solitary waves: a numerical study of the proximal femur.

This study investigates the suitability of a relatively new non-destructive evaluation (NDE) technique for the detection of non-visible defects in cellular solids using highly nonlinear solitary waves (HNSWs) in a one-dimensional granular chain. Specifically, the HNSW-based NDE approach is employed to identify the existence of micro-fractures in trabecular bone within the femoral neck (FN) and the intertrochanteric (IT) region of the proximal femur which are fracture-prone sites due to their relatively low bone density, particularly in osteoporosis patients. The availability of a HNSW-based bone quality assessment tool could not only help in early diagnosis of osteoporosis but also affect surgical decisions and improve clinical outcomes in joint replacement surgeries which motivated this study. To obtain a realistic representation of the trabecular microstructure, high-resolution finite-element (FE) models of the FN and the IT region are first constructed using a topology optimization-based bone reconstruction scheme. Then, artificial defects in the form of fractured ligaments are generated in the FN and IT models by selectively disconnecting various struts within the trabecular network. Using the FE models as the inspection medium, hybrid discrete-element/finite-element (DE/FE) simulations are performed to examine the interaction of the HNSWs with the cellular bone samples through two different inspection modes, i.e., inspection via direct contact with the sample and indirect contact through an adequately chosen face sheet inserted between the cellular sample and the granular chain. The delays and amplitudes of the HNSWs are used to estimate the effective elastic moduli of the cellular samples and these estimates were found to be reasonably accurate only in case the face sheet was applied. For the latter case, it was shown that the HNSW-based modulus estimates can be used as indicators for defect detection, allowing to discern between pristine and damaged cellular solids. These results suggest that HNSW-based NDE is a reliable and cost-effective technique for the identification of defects in cellular solids, and is expected to find applications in various fields, such as non-invasive screening of bone diseases and fractures, or damage detection in additively manufactured cellular structures.

Direct observation of edge modes in zigzag granular chains

As a new class of artificial elastic materials, granular crystals are mechanical structures of elastic beads arranged in contact through a lattice. One important feature of wave dynamics in granular crystals is that it highly relies on the contact mechanics, allowing for exotic wave transport properties such as rotational waves, solitary waves, slow edge waves, topological edge waves, etc. Realizing granular structures with well-predicted wave physics not only renders these new properties to mechanical systems, but provides also significant possibilities for advanced elastic wave control scenarios. Here, we theoretically and experimentally study the linear wave dynamics in one-dimensional (1D) zigzag granular chains constructed with macroscopic spherical stainless steel/tungsten beads. A spring–mass model including normal, shear and bending mechanical couplings between beads is proposed to characterize the wave dynamics in the chain, which turns out to exhibit remarkable agreement with the experimental measurements. Our work confirms the existence of localized translational–rotational coupled modes at the ends of granular chains, and it might motivate future studies for novel topological wave effects in granular structures.

Predictions of the elastic modulus of trabecular bone in the femoral head and the intertrochanter: a solitary wave-based approach.

This paper deals with the numerical prediction of the elastic modulus of trabecular bone in the femoral head (FH) and the intertrochanteric (IT) region via site-specific bone quality assessment using solitary waves in a one-dimensional granular chain. For accurate evaluation of bone quality, high-resolution finite element models of bone microstructures in both FH and IT are generated using a topology optimization-based bone microstructure reconstruction scheme. A hybrid discrete element/finite element (DE/FE) model is then developed to study the interaction of highly nonlinear solitary waves in a granular chain with the generated bone microstructures. For more robust and reliable prediction of the bone's mechanical properties, a face sheet is placed at the interface between the last chain particle and thebone microstructure, allowing more bone volume to be engaged in the dynamic deformation during interaction with the solitary wave. The hybrid DE/FE model was used to predict the elastic modulus of the IT and FH by analysing the characteristic features of the two primary reflected solitary waves. It was found that the solitary wave interaction is highly sensitive to the elastic modulus of the bone microstructure and can be used to identify differences in bone density. Moreover, it was found that the use of a relatively stiff face sheet significantly reduces the sensitivity of the wave interaction to local stiffness variations across the test surface of the bone, thereby enhancing the robustness and reliability of the proposed method. We also studied the effect of the face sheet thickness on the characteristics of the reflected solitary waves and found that the optimal thickness that minimizes the error in the modulus predictions is 4mm for the FH and 2mm for the IT, if the primary reflected solitary wave is considered in the evaluation process. We envisage that the proposed diagnostic scheme, in conjunction with 3D-printed high-resolution bone models of an actual patient, could provide a viable solution to current limitations in site-specific bone quality assessment.

Coupling mechanism of highly nonlinear solitary waves with damaged composite material plates

The coupling interaction between nonlinear solitary waves in one-dimensional granular chains and damaged composite material plates is considered. Based on Hertz contact law and meso-mechanical model of stiffness reduction of composite material plates when the fiber breakage is the main damage mode, the coupled differential equations of particle chains and damaged composite material plates are derived. By solving the differential equations with Runge–Kutta method to get the velocity and displacement curves of particles and analyzing the delays and amplitude ratios of reflected waves, it is found that the damage quantity, fiber volume fraction, and thickness of damaged composite material plates as well as gravity have an effect on solitary waves. The preliminary research results provide a theoretical basis for nondestructive testing of damaged composite material plates by using solitary waves.

Quantitatively solitary wave tuning strategies based on one-dimensional cylindrical granular chains

One-dimensional (1D) granular systems support solitary wave propagation; however, solitary wave tuning, which serves as an effective strategy in stress wave control to achieve highly intelligent and efficient wave attenuation, still lacks. Herein, we design a 1D cylindrical granular system and comprehensively investigate solitary wave tuning strategies based on the system through mass, modulus, and thickness mismatch. We establish the experiment setups by using a vertical sliding track to observe the basic solitary wave traveling behavior as well as to validate the numerical simulation model. By considering the three governing wave tuning strategies, i.e., mass, modulus, and thickness mismatch, we parametrically explore the relations between the compression ratio and the mismatch values of different indicators. Based on the simple single-layer 1D chain, we further design a multi-layered 1D chain where the middle layers serve as tuning layers such that both “strong–weak” and “weak–strong” interfaces are created. A full tuning range is obtained by coupling different strategies based on the multilayer granular chains. Particularly, for a single layer granular chain, a mechanism map is established based on a normalized Ashby plot and numerical results of compression ratios, providing insights into precisely designing new attenuation systems. Results unlock the unique solitary wave tuning mechanism and provide design guidance for next-generation signal measurement, detection, and monitoring system.

Universal design law of equivalent systems for Nesterenko solitary waves transmission

Wave propagation behavior within 1D granular chain has been a much-studied topic in the past decades due to its novel physical wave phenomena. The initial motivation of the present study was prompted by how to construct a granular system with desired wave properties and particle motions. Here, we systematically report the methodology to construct equivalent systems (systems with identical output) using theoretical analysis, experiments and numerical simulation. We investigate the properties of Nesterenko solitary wave supported by one-dimensional granular chains and achieve an equivalent wave transmission among various materials and dimensions. The results of typical experiments and finite element analysis fully support the established theoretical predictions based on Hertz contact and long wavelength approximation theory, which yields a broad class of equivalent systems. Finally, an instructive mechanism map, containing various equivalent systems, is proposed to clarify the relationship between geometric parameters and material properties, providing insights into designing a desired nonlinear dynamic system and guidance for potential engineering applications.

Nonlinear dynamics of coupled transverse-rotational waves in granular chains.

The nonlinear dynamics of coupled waves in one-dimensional granular chains with and without a substrate is theoretically studied accounting for quadratic nonlinearity. The multiple time scale method is used to derive the nonlinear dispersion relations for infinite granular chains and to obtain the wave solutions for semi-infinite systems. It is shown that the sum frequency and difference frequency components of the coupled transverse-rotational waves are generated due to their nonlinear interactions with the longitudinal wave. Nonlinear resonances are not present in the chain with no substrate where these frequency components have low amplitudes and exhibit beating oscillations. In the chain positioned on a substrate two types of nonlinear resonances are predicted. At resonance, the fundamental frequency wave amplitudes decrease and the generated frequency component amplitudes increase along the chain, accompanied by the oscillations due to the wave number asynchronism. The results confirm the possibility of a highly efficient energy transfer between the waves of different frequencies, which could find applications in the design of acoustic devices for energy transfer and energy rectification.

Influencing factors of the performance of an impact buffering made of the composite granular chain

The influencing factors of the performances of an impact buffering made of the composite granular chain of spheres were investigated by both the finite element analysis and experiments. The composite granular chain was prepared by inserting an adjustable chain into a one-dimensional steel granular chain of spheres. It is suggested to select the key influencing factor as λ=(EvEs)23ρvρs, which determined the features of output impulses. The amplitude ratios of primary and secondary output impulses to the incident impulse of highly nonlinear solitary wave (HNSW) and the time delay between the primary and secondary output impulses demonstrated the two-stage dependencies on the parameter of λ. The two stages were separated from each other by the case of λ = =1. It was suggested to construct an impact buffering with near-zero λ in order to minimize the amplitude of primary output impulse. The amplitude attenuation of the primary output impulse in the adjustable chain followed the power law and the power exponent was negatively correlated with the value of λ. As for a composite chain with a given parameter of λ, through increasing the length of the adjustable chain, the amplitude of the primary output impulse could be decreased in order to improve the performance of impact buffering.

Linear and nonlinear surface waves in two-dimensional, hexagonal close-packed granular crystals

Granular crystals (GCs) are ordered arrays of spherical particles in contact, which have been shown to exhibit rich nonlinear dynamics stemming from Hertzian contact interactions. They are typically modeled as discrete networks of rigid body spheres, which may undergo translational and rotational motion, with Hertzian interactions modeled as effective nonlinear normal and shear springs. Most of the existing literature on GCs concerns one-dimensional chains of spheres, though a growing body of recent work has explored two-dimensional systems in linear and nonlinear regimes, with multiple packing geometries. In a recent analytical study, Rayleigh-like surface waves were shown to exist in the linear regime for two-dimensional GCs with square packing [Phys. Rev. E 93, 023008 (2016)]. In the present work, we report on surface waves in two-dimensional granular crystals with hexagonal close-packed (HCP) geometry. We demonstrate analytically that, in the linear regime, HCP GCs support Rayleigh-like surface waves analogous to the square-packed case, as well as a leaky surface wave mode with quasi-longitudinal polarization. Extensions of the linear surface modes into the nonlinear regime are demonstrated via numerical examples and compared to results from one-dimensional granular chains. [Work supported by the postdoctoral program at ARL:UT.]

Simulation on propagation characteristics of solitary waves in a one-dimensional charged granular chain

Solitary waves propagating in a one dimensional charged granular chain has been investigated numerically. Electrically long-range interaction is introduced into such a nonlinear system for the first time. The propagating characteristics of the waves due to physical parameters of materials are studied systematically. It is found that both single solitary wave and multi-solitary wave can be excited in the granular chain. Numerical results show that the amplitude of the solitary wave decays exponentially because of damping as time increases even if the Coulomb’s force is taken into account. Specifically, Young’s modulus and charge quantity have slightly effect on the amplitude attenuation rate. However, the damping coefficient and the density of particles have significant effect.

Phase shift of solitary wave in a one-dimensional granular chain

In this paper, both the fifth-order Runge–Kutta numerical scheme and binary collision approximation are used to study the phase shift. Both numerical and theoretical results are shown that the solitary wave after head-on collision propagates along the chain behind the reference wave in both even and odd numbers of grain chains. It is the well-known feature of the appearance of the phase shift. Those results are in agreement with the experimental results. Furthermore, it is found that the phase shift is not only related to the collision position of the waves, but also to the position where the time is measured. The value of phase shift increases nonmonotonously with increasing the velocity of the opposite propagation of the wave. Binary collision approximation is applied to analyze the phase shift, and it is found that theoretical results agree well with numerical results, especially in the case of phase shift in odd chain.

Decaying solitary waves propagating in one-dimensional damped granular chain**Project supported by the National Natural Science Foundation of China (Grant Nos.11565021 and 11047010) and the Scientific Research Foundation of Northwest Normal University, China (Grant No. NWNU-LKQN-16-3).

We numerically investigate the nonlinear waves propagating in a one-dimensional particle chain when the damping effect is taken into account. It is found that decaying solitary waves exist, in which the amplitude of the wave decreases exponentially as time increases. Meanwhile, the velocity of the solitary wave also slows down as time goes. This result implies that the damping coefficient is an important parameter in such a nonlinear system. Theoretical analysis has also been done by the reductive perturbation method. The result indicates that the nonlinear waves propagating in such a system can be described by the damped KdV equation.

Elastic Wannier-Stark ladders and Bloch oscillations in 1D granular crystals

We report the numerical and experimental study of elastic Wannier-Stark ladders and Bloch oscillations in a tunable one-dimensional granular chain consisting of cylindrical particles. The Wannier-Stark ladders are obtained by tuning the contact angles to introduce a gradient in the contact stiffness along the granular chain. These ladders manifest as resonant modes localized in the space. When excited at the corresponding resonant frequencies, we demonstrate the existence of time-resolved Bloch oscillations. The direct velocity measurements using laser Doppler vibrometry agree well with the numerical simulation results. We also show the possibility of further tailoring these Bloch oscillations by numerical simulations. Such tunable systems could be useful for applications involving the spatial localization of elastic energy.

Tunable evolutions of shock absorption and energy partitioning in magnetic granular chains

In this paper, we investigate the tunable characteristics of shock waves propagating in one-dimensional magnetic granular chains at various chain lengths and magnetic flux densities. According to the Hertz contact theory and Maxwell principle, a discrete element model with coupling elastic and field-induced interaction potentials of adjacent magnetic grains is proposed. We also present hard-sphere approximation analysis to describe the energy partitioning features of magnetic granular chains. The results demonstrate that, for a fixed magnetic field strength, when the chain length is greater than two times of the wave width of the solitary wave, the chain length has little effect on the output energy of the system; for a fixed chain length, the shock absorption and energy partitioning features of magnetic granular chains are remarkably influenced by varying magnetic flux densities. This study implies that the magnetic granular chain is potential to construct adaptive shock absorption components for impulse mitigation.

A Novel Envelope Soliton Solution to the Granular Crystal Model**Supported by the National Natural Science Foundation of China under Grant Nos. 1604121 and 11464012

A theoretical work on envelope solitons to a one-dimensional granular chain model is reported. In the small amplitude approximation, we analytically solve the equation of motion with the help of the semidiscrete multiple-scale method. Our results show that the granular chain model can support an asymmetric high-order envelope soliton under the certain condition. It is found that the second-harmonic term of this high-order envelope soliton has an additional phase. In addition, the influence of both the material parameter and the static load on the localized features of the high-order envelope soliton is discussed.

Wave propagation in one-dimensional microscopic granular chains.

We employ noncontact optical techniques to generate and measure stress waves in uncompressed, one-dimensional microscopic granular chains, and support our experiments with discrete numerical simulations. We show that the wave propagation through dry particles (150 μm radius) is highly nonlinear and it is significantly influenced by the presence of defects (e.g., surface roughness, interparticle gaps, and misalignment). We derive an analytical relation between the group velocity and gap size, and define bounds for the formation of highly nonlinear solitary waves as a function of gap size and axial misalignment.

Shock wave in a one-dimensional granular chain under Hertz contact

The shock wave in one-dimensional bead chain is studied numerically. When the shock wave arrives, the bead velocity oscillates around the piston velocity. It is found that the shock front is composed of several solitary waves and the limitation of the maximum bead velocity is 2 times the piston velocity in the limiting case where the initial overlap is zero. If the initial overlap is not zero, then the maximum bead velocity is less than 2 times the piston velocity but larger than the piston velocity. As the initial overlap increases from zero to the finite value, the shock velocity depends on not only the piston velocity but also the initial overlap. The crossover of the dependence of the shock velocity on the piston velocity from the zero initial prestress to the finite value is obtained in the present manuscript. It is an improvement of the results presented in Phys. Rev. Lett. 108, 058001 (2012)10.1103/PhysRevLett.108.058001. In other words, the dependence of the shock velocity on the parameters of the granular materials is given.

Experimental Study of Nonlinear Resonances and Anti-Resonances in a Forced, Ordered Granular Chain

We experimentally study a one-dimensional uncompressed granular chain composed of a finite number of identical spherical elastic beads with Hertzian interactions. The chain is harmonically excited by an amplitude- and frequency-dependent boundary drive at its left end and has a fixed boundary at its right end. Such ordered granular media represent an interesting new class of nonlinear acoustic metamaterials, since they exhibit essentially nonlinear acoustics and have been designated as “sonic vacua” due to the fact that their corresponding speed of sound (as defined in classical acoustics) is zero. This paves the way for essentially nonlinear and energy-dependent acoustics with no counterparts in linear theory. We experimentally detect time-periodic, strongly nonlinear resonances whereby the particles (beads) of the granular chain respond at integer multiples of the excitation period, and which correspond to local peaks of the maximum transmitted force at the chain’s right, fixed end. In between these resonances we detect a local minimum of the maximum transmitted forces corresponding to an anti-resonance in the stationary-state dynamics. The experimental results of this work confirm previous theoretical predictions, and verify the existence of strongly nonlinear resonance responses in a system with a complete absence of any linear spectrum; as such, the experimentally detected nonlinear resonance spectrum is passively tunable with energy and sensitive to dissipative effects such as internal structural damping in the beads, and friction or plasticity effects. We compare the experimental results with direct numerical simulations of the granular network and deduce satisfactory agreement.