We report on the strongly nonlinear dynamics of an array of weakly coupled, noncompressed, parallel granular chains subject to a local initial impulse. The motion of the granules in each chain is constrained to be in one direction that coincides with the orientation of the chain. We show that in spite of the fact that the applied impulse is applied to one of the granular chains, the resulting pulse that initially propagates only in the excited chain gets gradually equipartitioned between its neighboring chains and eventually in all chains of the array. In particular, the initially strongly localized state of energy distribution evolves towards a final stationary state of formation of identical solitary waves that propagate in each one of the chains. These solitary waves are synchronized and have identical speeds. We show that the phenomenon of primary pulse equipartition between the weakly coupled granular chains can be fully reproduced in coupled binary models that constitute a significantly simpler model that captures the main qualitative features of the dynamics of the granular array. The results reported herein are of major practical significance since it indicates that the weakly coupled array of granular chains is a medium in which an initially localized excitation gets gradually defocused, resulting in drastic reduction of propagating pulses as they are equipartitioned among all chains.

References

References
1.
Nesterenko
,
V. F.
,
2001
,
Dynamics of Heterogeneous Materials
,
Springer-Verlag
, New York.
2.
Nesterenko
,
V. F.
,
1983
, “
Propagation of Nonlinear Compression Pulses in Granular Media
,”
J. Appl. Mech. Tech. Phy.
,
24
(
5
), pp.
733
743
.10.1007/BF00905892
3.
Sen
,
S.
,
Hong
,
J.
,
Bang
,
J.
,
Avalos
,
E.
, and
Doney
,
R.
,
2008
, “
Solitary Waves in the Granular Chain
,”
Phys. Rep.
,
462
(
2
), pp.
21
66
.10.1016/j.physrep.2007.10.007
4.
Daraio
,
C.
,
Nesterenko
,
V. F.
,
Herbold
,
E. B.
, and
Jin
,
S.
,
2006
, “
Tunability of Solitary Wave Properties in One-Dimensional Strongly Nonlinear Phononic Crystals
,”
Phys. Rev. E
,
73
(
2
), p.
026610
.10.1103/PhysRevE.73.026610
5.
Coste
,
C.
,
Falcon
,
E.
, and
Fauve
,
S.
,
1997
, “
Solitary Waves in a Chain of Beads Under Hertz Contact
,”
Phys. Rev. E
,
56
(
5
), pp.
6104
6117
.10.1103/PhysRevE.56.6104
6.
Job
,
S.
,
Melo
,
F.
,
Sokolow
,
A.
, and
Sen
,
S.
,
2005
, “
How Hertzian Solitary Waves Interact With Boundaries in a 1D Granular Medium
,”
Phys. Rev. Lett.
,
94
(
17
), p.
178002
.10.1103/PhysRevLett.94.178002
7.
Chatterjee
,
A.
,
1999
, “
Asymptotic Solution for Solitary Waves in a Chain of Elastic Spheres
,”
Phys. Rev. E
,
59
(
5
), pp.
5912
5919
.10.1103/PhysRevE.59.5912
8.
Lazaridi
,
A. N.
, and
Nesterenko
,
V. F.
,
1985
, “
Observation of a New Type of Solitary Waves in a One-Dimensional Granular Medium
,”
J. Appl. Mech. Tech. Phy.
,
26
(
3
), pp.
405
408
.10.1007/BF00910379
9.
Porter
,
M. A.
,
Daraio
,
C.
,
Herbold
,
E. B.
,
Szelengowicz
,
I.
, and
Kevrekidis
,
P. G.
,
2008
, “
Highly Nonlinear Solitary Waves in Periodic Dimer Granular Chains
,”
Phys. Rev. E
,
77
(
1
), p.
015601
.10.1103/PhysRevE.77.015601
10.
Herbold
,
E.
,
Kim
,
J.
,
Nesterenko
,
V.
,
Wang
,
S.
, and
Daraio
,
C.
,
2009
, “
Pulse Propagation in a Linear and Nonlinear Diatomic Periodic Chain: Effects of Acoustic Frequency Band-Gap
,”
Acta Mech.
,
205
(
1
), pp.
85
103
.10.1007/s00707-009-0163-6
11.
Doney
,
R.
, and
Sen
,
S.
,
2006
, “
Decorated, Tapered, and Highly Nonlinear Granular Chain
,”
Phys. Rev. Lett.
,
97
(
15
), p.
155502
.10.1103/PhysRevLett.97.155502
12.
Theocharis
,
G.
,
Boechler
,
N.
,
Kevrekidis
,
P. G.
,
Job
,
S.
,
Porter
,
M. A.
, and
Daraio
,
C.
,
2010
, “
Intrinsic Energy Localization Through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals
,”
Phys. Rev. E
,
82
(
5
), p.
056604
.10.1103/PhysRevE.82.056604
13.
Theocharis
,
G.
,
Kavousanakis
,
M.
,
Kevrekidis
,
P. G.
,
Daraio
,
C.
Porter
,
M. A.
, and
Kevrekidis
,
I. G.
,
2009
, “
Localized Breathing Modes in Granular Crystals With Defects
,”
Phys. Rev. E
,
80
(
6
), p.
066601
.10.1103/PhysRevE.80.066601
14.
Boechler
,
N.
,
Theocharis
,
G.
,
Job
,
S.
,
Kevrekidis
,
P. G.
,
Porter
,
M. A.
, and
Daraio
,
C.
,
2010
, “
Discrete Breathers in One-Dimensional Diatomic Granular Crystals
,”
Phys. Rev. Lett.
,
104
(
24
), p.
244302
.10.1103/PhysRevLett.104.244302
15.
Starosvetsky
,
Y.
,
Jayaprakash
,
K. R.
, and
Vakakis
,
A. F.
,
2012
, “
Scattering of Solitary Waves and Excitation of Transient Breathers in Granular Media by Light Intruders and No Precompression
,”
ASME J. Appl. Mech.
,
79
(
1
), p.
011001
.10.1115/1.4003360
16.
Jayaprakash
,
K. R.
,
Starosvetsky
,
Y.
, and
Vakakis
,
A. F.
,
2011
, “
New Family of Solitary Waves in Granular Dimer Chains With No Precompression
,”
Phys. Rev. E
,
83
(
3
), p.
036606
.10.1103/PhysRevE.83.036606
17.
Hong
,
J.
, and
Xu
,
A.
,
2002
, “
Nondestructive Identification of Impurities in Granular Medium
,”
Appl. Phys. Lett.
,
81
(
25
), pp.
4868
4870
.10.1063/1.1522829
18.
Hong
,
J.
,
2005
, “
Universal Power-Law Decay of the Impulse Energy in Granular Protectors
,”
Phys. Rev. Lett.
,
94
(
10
), p.
108001
.10.1103/PhysRevLett.94.108001
19.
MacKay
,
R.
,
1999
, “
Solitary Waves in a Chain of Beads Under Hertz Contact
,”
Phys. Lett. A
,
251
(
3
), pp.
191
192
.10.1016/S0375-9601(98)00867-6
20.
Ji
,
J.-Y.
, and
Hong
,
J.
,
1999
, “
Existence Criterion of Solitary Waves in a Chain of Grains
,”
Phys. Lett. A
,
260
(
1–2
), pp.
60
61
.10.1016/S0375-9601(99)00488-0
21.
Friesecke
,
G.
, and
Wattis
,
J. A. D.
,
1994
, “
Existence Theorem for Solitary Waves on Lattices
,”
Commun. Math. Phys.
,
161
(
2
), pp.
391
418
.10.1007/BF02099784
22.
Starosvetsky
,
Y.
, and
Vakakis
,
A. F.
,
2010
, “
Traveling Waves and Localized Modes in One-Dimensional Homogeneous Granular Chains With No Precompression
,”
Phys. Rev. E
,
82
(
2
), p.
026603
.10.1103/PhysRevE.82.026603
23.
Jayaprakash
,
K.
,
Starosvetsky
,
Y.
,
Vakakis
,
A.
,
Peeters
,
M.
, and
Kerschen
,
G.
,
2011
, “
Nonlinear Normal Modes and Band Zones in Granular Chains With No Precompression
,”
Nonlinear Dyn.
,
63
(
3
), pp.
359
385
.10.1007/s11071-010-9809-0
24.
Starosvetsky
,
Y.
,
Hasan
,
M. A.
,
Vakakis
,
A. F.
, and
Manevitch
,
L. I.
,
2012
, “
Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations
,”
SIAM J. Appl. Math.
,
72
(
1
), p.
337
361
.10.1137/110835128
25.
Rosas
,
A.
, and
Lindenberg
,
K.
,
2004
, “
Pulse Velocity in a Granular Chain
,”
Phys. Rev. E
,
69
(
3
), p.
037601
.10.1103/PhysRevE.69.037601
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