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

Potekin, Randi; McFarland, D. Michael; Vakakis, Alexander F.

doi:10.1007/s10035-016-0657-6

Cited by 7 publications

(6 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, to simplify the mathematical model we assume that all granular motions are planar; in addition, assuming that the intermediate elastic solid is sufficiently thin we consider the plane stress approximation for the equations of infinitesimal linear elasticity governing its dynamics, so the that its corresponding deformations are also planar. Following the models in the previous works [23,24], the discrete element (DE) method is applied to model the acoustics of the left and right granular media, and the finite element (FE) method is applied to model the elastic solid. The two computational models are decoupled by accurately computing the interaction forces that couple the discrete (granular) and continuum (elastic solid) components of the interface at successive time steps [23,24].…”

Section: System Descriptionmentioning

confidence: 99%

“…This requires the study the ordered granular media with non-standard flexible boundary conditions. Potekin et al [23] first developed an algorithm to study wave scattering at the interface of a 1D granular chain with a linearly elastic cord (tensioned wire), which incorporated interrelated iterations and interpolations at successive time steps. Zhang et al [24] extended that algorithm to study nonlinear wave scattering at the interface of a 1D granular chain with a linear membrane.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, accounting for these effects may significantly improve the agreement between numerical simulations and experiments [12,26]. By incorporating the combined effects of friction and the flexible boundaries, Wang et al [29] extended the algorithm developed by Potekin et al [23] and Zhang et al [24] to 2D granularsolid interfaces consisting of closely packed, hexagonally ordered granules in contact with thin plates. To avoid numerical instabilities due to frictional effects, a smooth approximation was considered instead of the discontinuous Coulomb friction forces, and a convergence criterion based on the eigenvalues of a linearized iterative map of the granule-plate interaction forces at each time step was formulated to ensure the stability and robustness of the algorithm [27].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry

2021

View full text Add to dashboard Cite

We study non-reciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium. Specifically, a left, larger-scale and a right smaller-scale granular media composed of two-dimensional (2D), initially uncompressed hexagonally packed granules are interfacing with an intermediate linearly elastic solid, modeled either as a thin elastic plate or a linear Euler-Bernoulli beam. The granular media are modeled by discreteelements and the elastic solid by finite-elements assuming a plane stress approximation for the thin plate. Accounting for the combined effects of Hertzian, frictional and rotational interactions in the granular media, as well as the highly discontinuous interfacial effects between the (discrete) granular media and the (continuous) intermediate elastic solid, the nonlinear acoustics of the integrated system is computationally studied subject to a half-sine shock excitation applied to a boundary granule of either the left or right granular medium. The highly discontinuous and nonlinear interaction forces coupling the granular media to the elastic solid are accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps. Numerical convergence is ensured by monitoring the (linearized) eigenvalues of a nonlinear map of interface forces at each (variable) time step. Due to the strong nonlinearity and hierarchical asymmetry of the left and right granular media, time scale disparity occurs in the response of the interface which breaks acoustic reciprocity. Specifically, depending on the location and intensity of the applied shock, propagating wavefronts are excited in the granular media, which, in turn, excite either (slow) low-frequency vibrations or (fast) highfrequency acoustics in the intermediate elastic medium. This scale disparity is due to the size disparity of the left and right granular media, which yields drastically different wave speeds in the resulting propagating wavefronts. As a result, the continuum part of the interface responds with either low-frequency vibrationswhen the shock is applied to the larger-scale granular medium, or high-frequency waveswhen the shock is applied to the smaller-scale granular medium. This provides the fundamental mechanism for breaking reciprocity in the interface. The non-reciprocal interfacial acoustics studied here apply to a broad class of asymmetric hybrid (discrete-continuum) nonlinear systems, and can inform predictive designs of highly effective granular shock protectors or granular acoustic diodes.

show abstract

Section: System Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry

2021

View full text Add to dashboard Cite

show abstract

“…The dynamic wave propagation in a granular chain can be argued to be a Markovian process; the initial waveform (displacement/velocity of the particles) and the granular chain properties (pre-compression, sizes/masses of the particles through which the mechanical wave propagates) are sufficient to construct/predict successively the waveform at later time intervals [ 43 , 44 , 45 ]. The transition probability functions of the Markovian processes can be written in the form of the Chapman–Kolmogorov equation, one of the versions of this equation is the master Equation [ 46 ].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Model for Energy Propagation in Disordered Granular Chains

2021

View full text Add to dashboard Cite

Energy transfer is one of the essentials of mechanical wave propagation (along with momentum transport). Here, it is studied in disordered one-dimensional model systems mimicking force-chains in real systems. The pre-stressed random masses (other types of disorder lead to qualitatively similar behavior) interact through (linearized) Hertzian repulsive forces, which allows solving the deterministic problem analytically. The main goal, a simpler, faster stochastic model for energy propagation, is presented in the second part, after the basic equations are re-visited and the phenomenology of pulse propagation in disordered granular chains is reviewed. First, the propagation of energy in space is studied. With increasing disorder (quantified by the standard deviation of the random mass distribution), the attenuation of pulsed signals increases, transiting from ballistic propagation (in ordered systems) towards diffusive-like characteristics, due to energy localization at the source. Second, the evolution of energy in time by transfer across wavenumbers is examined, using the standing wave initial conditions of all wavenumbers. Again, the decay of energy (both the rate and amount) increases with disorder, as well as with the wavenumber. The dispersive ballistic transport in ordered systems transits to low-pass filtering, due to disorder, where localization of energy occurs at the lowest masses in the chain. Instead of dealing with the too many degrees of freedom or only with the lowest of all the many eigenmodes of the system, we propose a stochastic master equation approach with reduced complexity, where all frequencies/energies are grouped into bands. The mean field stochastic model, the matrix of energy-transfer probabilities between bands, is calibrated from the deterministic analytical solutions by ensemble averaging various band-to-band transfer situations for short times, as well as considering the basis energy levels (decaying with the wavenumber increasing) that are not transferred. Finally, the propagation of energy in the wavenumber space at transient times validates the stochastic model, suggesting applications in wave analysis for non-destructive testing, underground resource exploration, etc.

show abstract

“…The dynamic wave propagation in a granular chain can be argued as a Markovian process, the initial waveform (displacement/velocity of the particles) and the granular chain properties (pre-compression, sizes/masses of the particles through which the mechanical wave propagates) are sufficient to construct/predict successively the waveform at later time intervals [87,195,213]. The transition probability functions of the Markovian processes can be written in the form of Chapman-Kolmogrov equation, one of the versions of this equation is the Master Equation [103].…”

Section: Introductionmentioning

confidence: 99%

Elasticity and Wave Propagation in Granular Materials

Taghizadeh¹

View full text Add to dashboard Cite

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

Cited by 7 publications

References 27 publications

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry

Stochastic Model for Energy Propagation in Disordered Granular Chains

Elasticity and Wave Propagation in Granular Materials

Contact Info

Product

Resources

About