TY - JOUR
T1 - Solitary waves in a general class of granular dimer chains
AU - Jayaprakash, K. R.
AU - Vakakis, Alexander F.
AU - Starosvetsky, Yuli
N1 - Funding Information:
This work was funded by MURI grant US ARO W911NF-09-1-0436, Dr. David Stepp is the grant monitor. The authors acknowledge this support.
PY - 2012/8
Y1 - 2012/8
N2 - We report on a countable infinity of traveling solitary waves in a class of highly heterogeneous ordered one-dimensional granular media, in particular, granular dimers composed of an infinite number of periodic sets of heavy elastic spherical beads in contact with N light ones; these media are denoted as 1: N granular dimers. Perfectly elastic Hertzian interaction between beads is assumed and no dissipative forces are taken into account in our study; moreover, zero pre-compression is assumed, rendering the dynamics strongly nonlinear through complete elimination of linear acoustics from the problem. After developing a general asymptotic methodology for the 1: N granular dimer, we focus on the case N 2 and prove numerically and asymptotically the existence of a countable infinity of traveling solitary waves in the 1:2 dimer chain. These solitary waves, which can be regarded as anti-resonances in these strongly nonlinear media, are found to be qualitatively different than those previously studied in homogeneous and 1:1 dimer chains (i.e., composed of alternating heavy and light beads) which possess symmetric velocity waveforms. In contrast, for traveling solitary waves in 1:2 dimers, the velocity waveforms of the responses of the heavy beads are symmetric, whereas those of the light beads are non-symmetric. Interestingly, we show that no such solitary waves can be realized in general 1: N granular dimers with N > 2, although near-solitary waves can exist in these systems based on slow-fast frequency approximations.
AB - We report on a countable infinity of traveling solitary waves in a class of highly heterogeneous ordered one-dimensional granular media, in particular, granular dimers composed of an infinite number of periodic sets of heavy elastic spherical beads in contact with N light ones; these media are denoted as 1: N granular dimers. Perfectly elastic Hertzian interaction between beads is assumed and no dissipative forces are taken into account in our study; moreover, zero pre-compression is assumed, rendering the dynamics strongly nonlinear through complete elimination of linear acoustics from the problem. After developing a general asymptotic methodology for the 1: N granular dimer, we focus on the case N 2 and prove numerically and asymptotically the existence of a countable infinity of traveling solitary waves in the 1:2 dimer chain. These solitary waves, which can be regarded as anti-resonances in these strongly nonlinear media, are found to be qualitatively different than those previously studied in homogeneous and 1:1 dimer chains (i.e., composed of alternating heavy and light beads) which possess symmetric velocity waveforms. In contrast, for traveling solitary waves in 1:2 dimers, the velocity waveforms of the responses of the heavy beads are symmetric, whereas those of the light beads are non-symmetric. Interestingly, we show that no such solitary waves can be realized in general 1: N granular dimers with N > 2, although near-solitary waves can exist in these systems based on slow-fast frequency approximations.
UR - http://www.scopus.com/inward/record.url?scp=84865249730&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84865249730&partnerID=8YFLogxK
U2 - 10.1063/1.4740060
DO - 10.1063/1.4740060
M3 - Article
AN - SCOPUS:84865249730
VL - 112
JO - Journal of Applied Physics
JF - Journal of Applied Physics
SN - 0021-8979
IS - 3
M1 - 034908
ER -