Interaction of traveling waves with mass-with-mass defects within a Hertzian chain

Kevrekidis, P. G.; Vainchtein, Anna; Garcia, M. Serra; Daraio, Chiara

doi:10.1103/physreve.87.042911

Cited by 38 publications

(34 citation statements)

References 38 publications

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“…Here and in what follows, n ranges from −N to N . For the analysis purposes it is convenient to rescale the governing equations (1) and the initial conditions (2) by introducing dimensionless displacementsū n ,v 0 and timet related to the original variables via [18,21]…”

Section: The Modelmentioning

confidence: 99%

Lattices with internal resonator defects

Hauver

Mei

et al. 2018

Phys. Rev. E

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We consider a variety of settings involving chains with one or more defects stemming from the introduction of nodes bearing internal resonators. Motivated by experimental results in woodpile elastic lattices with one or two defects, we consider a variety of different theoretical scenarios. These include multi-defect chains and their ability to transmit, reflect, and especially trap energy; they also include settings with linear vs. nonlinear defects of variable interaction exponent. Moreover, they involve defects which are spatially separated and either statically, or more effectively dynamically, enable the confinement of energy between the separated defects. Wherever possible, comparisons of the experiments with numerical simulations, as well as with theoretical intuition are also offered, to provide a justification for the observed findings.

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Section: The Modelmentioning

confidence: 99%

Lattices with internal resonator defects

Hauver

Mei

et al. 2018

Phys. Rev. E

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“…Admittedly, both of these realizations were chiefly linear (in the presence of externally imposed pre-compression of the chain) and aiming to illustrate the remarkable meta-material type properties that these systems possess. Yet, while a MwM nonlinear prototype was also demonstrated in [61], it was a different type of experiment that very recently realized highly nonlinear propagation in a locally resonant granular system [62]. In particular, the experiment of [62] built a so-called woodpile configuration consisting of orthogonally stacked rods (every second rod is aligned, in this alternating 0-90 degree configuration) and demonstrated that the internal vibrations of the rods can play the role of the local resonator within the granular chain.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves and their tails in locally resonant granular systems

Kevrekidis

Stefanov

2015

J. Phys. A: Math. Theor.

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In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, which will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of antiresonance conditions is identified for which solutions with genuinely rapidly decaying tails arise.

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“…An additional possibility which would avoid the complications (slow tail decay in Fourier space) of the Fourier calculation proposed above would be to revert (37) and (38) into real space. That would reduce the number of modes needed for an accurate computation of the solution.…”

Section: A the Pego-english Approach And The Existence Of Nanopteramentioning

confidence: 99%

“…A typical example of the result of the iteration scheme of (37) and (38) is given in Figures 14 and 15 for the case of ν = 0.05. Figure 14 illustrates the result of the iterative scheme of the equations towards the acquisition of a traveling wave solution.…”

Section: A the Pego-english Approach And The Existence Of Nanopteramentioning

confidence: 99%

Wave propagation in a strongly nonlinear locally resonant granular crystal

Vorotnikov

Starosvetsky

Theocharis

et al. 2018

Physica D: Nonlinear Phenomena

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In this work, we study the wave propagation in a recently proposed acoustic structure, the locally resonant granular crystal. This structure is composed of a one-dimensional granular crystal of hollow spherical particles in contact, containing linear resonators. The relevant model is presented and examined through a combination of analytical approximations (based on ODE and nonlinear map analysis) and of numerical results. The generic dynamics of the system involves a degradation of the well-known traveling pulse of the standard Hertzian chain of elastic beads. Nevertheless, the present system is richer, in that as the primary pulse decays, secondary ones emerge and eventually interfere with it creating modulated wavetrains. Remarkably, upon suitable choices of parameters, this interference "distills" a weakly nonlocal solitary wave (a "nanopteron"). This motivates the consideration of such nonlinear structures through a separate Fourier space technique, whose results suggest the existence of such entities not only with a single-side tail, but also with periodic tails on both ends. These tails are found to oscillate with the intrinsic oscillation frequency of the out-of-phase motion between the outer hollow bead and its internal linear attachment.

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Interaction of traveling waves with mass-with-mass defects within a Hertzian chain

Cited by 38 publications

References 38 publications

Lattices with internal resonator defects

Lattices with internal resonator defects

Traveling waves and their tails in locally resonant granular systems

Wave propagation in a strongly nonlinear locally resonant granular crystal

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