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Research Papers

Scattering of Solitary Waves and Excitation of Transient Breathers in Granular Media by Light Intruders and No Precompression

[+] Author and Article Information
Yuli Starosvetsky

Department of Mechanical Science and Engineering,  University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61822staryuli@illinois.edu

K. R. Jayaprakash

Department of Mechanical Science and Engineering,  University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61822kalkunt1@illinois.edu

Alexander F. Vakakis

Department of Mechanical Science and Engineering,  University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61822avakakis@illinois.edu

J. Appl. Mech 79(1), 011001 (Nov 14, 2011) (12 pages) doi:10.1115/1.4003360 History: Received November 24, 2009; Revised May 02, 2010; Posted January 05, 2011; Published November 14, 2011; Online November 14, 2011

We analyze the dynamics of strongly nonlinear granular chains of beads in Hertzian contact with light intruders. We show that the interactions of the light intruders with solitary pulses propagating through the granular medium can be approximately studied by reduced models of the intruders with only their neighboring beads under similar excitation conditions. Studying the reduced models, we identify weakly and strongly nonlinear regimes in the dynamics, depending on the degree of compression between beads and on the occurrence of separation between neighboring beads leading to collisions. We analyze weakly and strongly nonlinear oscillatory regimes of the intruder dynamics by multiple-scale analysis, and by applying special nonsmooth coordinate transformations. When separation between beads occurs, localized transient breathers are excited, corresponding to repeated collisions of an intruder with its neighbors. This leads to high-frequency scattering energy, and to radiation of energy in the granular medium in the form of low-amplitude slowly modulated oscillatory pulses. We find that repeated excitation of localized transient breathers by an array of periodically placed intruders can result in drastic reduction of the amplitude of a solitary wave propagating through the granular medium. This indicates that this type of granular media can be designed as effective shock attenuators.

Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Granular chain with a light mass intruder

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Figure 2

Excitation of the light intruder with 5% mass compared with the beads of the granular chain, showing the two distinct modes of intruder-chain interaction

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Figure 3

Comparison of the dynamics of reduced order models and the chain-intruder system for the first mode of interaction; T is the interval between subsequent bead separations and provides a characteristic time of the slow dynamics: (a) time series comparisons for the three-bead reduced model, (b) configurations of the reduced order models with different numbers of beads, (c) energy transfer across the intruder, (d) momentum transfer to the bead on the right of the intruder, and (e) combined momentum transfer to the beads on the right of the intruder

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Figure 4

Slowly varying frequency governing the linearized fast dynamics of the intruder

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Figure 5

Comparison of asymptotic results (------) with numerical simulations ——— for weakly nonlinear fast oscillations of the light intruder, with ɛ=0.01 and zero initial conditions except for v10=10

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Figure 6

Comparison of asymptotic results (------) with numerical simulations ——— for weakly nonlinear fast oscillations of the light intruder, with ɛ=0.10 and zero initial conditions except for v10=10

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Figure 7

Comparison of asymptotic results (------) with numerical simulations ——— for weakly nonlinear fast oscillations of the light intruder in the limit of the asymptotic analysis for ɛ=ɛcr=0.8251 and zero initial conditions except for v10=10

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Figure 8

Nonsmooth transformations for the fast time scale

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Figure 9

Comparison of asymptotic results (------) with numerical simulations ——— for strongly nonlinear oscillations of the light intruder, with ɛ=0.001 and initial conditions u10=2,v10=1,u20=1,v20=65.82,u30=0,v30=0

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Figure 10

Transient breathers in granular chains with a single light intruder of mass: (a) ɛ=0.0001, (b) ɛ=0.01, and (c) ɛ=0.5, (------) intruder and ——— neighboring beads (the two modes of intruder-chain interaction are indicated by labels 1 and 2)

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Figure 11

Velocity of the intruder for ɛ=0.01 during the formation of the transient breather: (a) time series and (b) wavelet spectrum

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Figure 12

One-dimensional granular chains with periodically placed light intruders every five regular beads

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Figure 13

Space-time plot of the maximum instantaneous bead energy of the impulsively forced granular chain with periodically spaced light intruders and free-free boundary conditions; the intensity of the scale of the data ranges from black (normalized energy level is equal to unity) to white (normalized energy level is equal to zero)

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Figure 14

Velocity profiles at selected intruder locations in the granular chain with an array of periodically spaced light intruders and free-free boundary conditions

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Figure 15

Space-time plot of the maximum instantaneous bead energy of the impulsively forced granular chain with periodically spaced light intruders and fixed-fixed boundary conditions

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Figure 16

Velocity profiles at selected intruder locations in the granular chain with an array of periodically spaced light intruders and fixed-fixed boundary conditions

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Figure 17

Velocity time series and corresponding wavelet spectrum of (a) the intruder at site 144 and (b) its neighbor regular bead at site 147

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