Journal
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
Volume 48, Issue 19, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1751-8113/48/19/195204
Keywords
mass in mass; locally resonant; traveling wave; mass with mass; Fourier transform; granular chain
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Funding
- National Science Foundation [DMS-1312856]
- ERC
- US-AFOSR [FA9550-12-10332]
- Binational (US-Israel) Science Foundation [2010239]
- US Department of Energy
- NSF [1313107]
- FP7-People [605096]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1313107] Funding Source: National Science Foundation
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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 anti-resonance conditions is identified for which solutions with genuinely rapidly decaying tails arise.
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