We study the transient responses of linear and nonlinear semi-infinite periodic media on linear elastic foundations under suddenly applied, high-frequency harmonic excitations. We show that “dynamic overshoot” phenomena are realized whereby, due to the high-rate of application of the high-frequency excitations, coherent traveling responses are propagating to the far fields of these media; and this, despite the fact that the high frequencies of the suddenly applied excitations lie well within the stop bands of these systems. For the case of a linear one-dimensional (1D) spring-mass lattice, a leading-order asymptotic approximation in the high frequency limit of the suddenly applied harmonic excitation shows that the transient dynamic overshoot is expressed in terms of the Green's function at its free end. Then, a two-dimensional (2D) strongly nonlinear granular network is considered, composed of two semi-infinite, ordered homogeneous granular lattices mounted on linear elastic foundations and coupled by weak linear coupling terms. A high-frequency harmonic excitation is applied to one of the granular lattices—designated as the “excited lattice”, with the other lattice designated as the “absorbing” one. The resulting dynamic overshoot phenomenon consists of a “pure” traveling breather, i.e., of a single propagating oscillatory wavepacket with a localized envelope, resulting from the balance of discreteness, dispersion, and strong nonlinearity. The pure breather is asymptotically studied by a complexification/averaging technique, showing nearly complete but reversible energy exchanges between the excited and absorbing lattices as the breather propagates to the far field. Verification of the analytical approximations with direct numerical simulations is performed.

References

References
1.
Hussein
,
M. I.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2014
, “
Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook
,”
ASME Appl. Mech. Rev.
,
66
(
4
), p.
040802
.
2.
Brillouin
,
L.
,
1953
,
Wave Propagation in Periodic Structures
,
Dover Publication
,
New York.
3.
Mead
,
D.
,
1975
, “
Wave Propagation and Natural Modes in Periodic Systems: I Mono-Coupled Systems
,”
J. Sound Vib.
,
40
(
1
), pp.
1
18
.
4.
Vakakis
,
A.
,
Gendelman
,
O.
,
Bergman
,
L.
,
McFarland
,
D.
,
Kerschen
,
G.
, and
Lee
,
Y.
,
2008
,
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems: I
,
Springer Verlag
, Dordrecht, The Netherlands.
5.
Vakakis
,
A.
,
Gendelman
,
O.
,
Bergman
,
L.
,
McFarland
,
D.
,
Kerschen
,
G.
, and
Lee
,
Y.
,
2008
,
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems: II
,
Springer Verlag
, Dorcrecht, The Netherlands.
6.
Vakakis
,
A. F.
, and
King
,
M. E.
,
1995
, “
Nonlinear Wave Transmission in a Mono-Coupled Elastic Periodic System
,”
J. Acoust. Soc. Am.
,
98
(
3
), pp.
1534
1546
.
7.
Vakakis
,
A. F.
, and
King
,
M. E.
,
1997
, “
Resonant Oscillations of a Weakly Coupled Nonlinear Layered System
,”
Acta Mech.
,
128
(
1–2
), pp.
59
80
.
8.
Boechler
,
N.
, and
Daraio
,
C.
,
2009
, “
An Experimental Investigation of Acoustic Band Gaps and Localization in Granular Elastic Chains
,”
ASME
Paper No. DETC2009-87427.
9.
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.
Nesterenko
,
V.
,
1983
, “
Propagation of Nonlinear Compression Pulses in Granular Media
,”
J. Appl. Mech. Tech. Phys.
,
24
(
5
), pp.
733
743
.
11.
Nesterenko
,
V.
, and
Lazaridi
,
A.
,
1985
, “
Observation of a New Type of Solitary Waves in a One-Dimensional Granular Medium
,”
J. Appl. Mech. Tech. Phys.
,
26
(
3
), pp.
405
408
.
12.
Nesterenko
,
V.
,
2001
,
Dynamics of Heterogeneous Materials
,
Springer Verlag
, New York.
13.
Daraio
,
C.
,
Nesterenko
,
V.
,
Herbold
,
B.
, and
Jin
,
S.
,
2006
, “
Tunability of Solitary Wave Properties in One-Dimensional Strongly Nonlinear Phononic Crystals
,”
Phys. Rev. E
,
73
(
2
), p.
026610
.
14.
Boechler
,
N.
,
Theocharis
,
G.
,
Job
,
S.
,
Kevrekidis
,
P.
,
Porter
,
M.
, and
Daraio
,
C.
,
2010
, “
Discrete Breathers in One-Dimensional Diatomic Granular Crystals
,”
Phys. Rev. Lett.
,
104
(
24
), p.
244302
.
15.
Starosvetsky
,
Y.
, and
Vakakis
,
A.
,
2010
, “
Traveling Waves and Localized Modes in One-Dimensional Homogeneous Granular Chains With No Pre-Compression
,”
Phys. Rev. E
,
82
(
2
), p.
026603
.
16.
Zhang
,
Y.
,
Hasan
,
M.
,
Starosvetsky
,
Y.
,
McFarland
,
D.
, and
Vakakis
,
A.
,
2015
, “
Nonlinear Mixed Solitary—Shear Waves and Pulse Equi-Partition in a Granular Network
,”
Physica D
,
291
, pp.
45
61
.
17.
Pozharskiy
,
D.
,
Zhang
,
Y.
,
Williams
,
M.
,
McFarland
,
D.
,
Kevrekidis
,
P.
,
Vakakis
,
A.
, and
Kevrekidis
,
I.
,
2015
, “
Nonlinear Resonances and Antiresonances of a Forced Sonic Vacuum
,”
Phys. Rev. E
,
92
(
6
), p.
063203
.
18.
Jayaprakash
,
K.
,
Starosvetsky
,
Y.
, and
Vakakis
,
A.
,
2011
, “
A New Family of Solitary Waves in Granular Dimmer Chains With No Pre-Compression
,”
Phys. Rev. E
,
83
(
3
), p.
036606
.
19.
Hasan
,
M.
,
Vakakis
,
A.
, and
McFarland
,
D.
,
2016
, “
Nonlinear Localization, Passive Wave Arrest and Traveling Breathers in Two-Dimensional Granular Networks With Discontinuous Lateral Boundary Conditions
,”
Wave Motion
,
60
, pp.
196
219
.
20.
Hasan
,
M.
,
Cho
,
S.
,
Remick
,
K.
,
Vakakis
,
A.
,
McFarland
,
D.
, and
Kriven
,
W.
,
2015
, “
Experimental Study of Nonlinear Acoustic Bands and Propagating Breathers in Ordered Granular Media Embedded in Matrix
,”
Granular Matter
,
17
(
1
), pp.
49
72
.
21.
James
,
G.
,
2011
, “
Nonlinear Waves in Newton's Cradle and the Discrete p-Schrödinger Equation
,”
Math. Models Methods Appl. Sci.
,
21
(
11
), pp.
2335
2377
.
22.
Starosvetsky
,
Y.
,
Hasan
,
M.
,
Vakakis
,
A.
, and
Manevitch
,
L.
,
2012
, “
Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations
,”
SIAM J. Appl. Math.
,
72
(
1
), pp.
337
361
.
23.
James
,
G.
,
Kevrekidis
,
P.
, and
Cuevas
,
J.
,
2013
, “
Breathers in Oscillator Chains With Hertzian Interactions
,”
Physica D
,
251
, pp.
39
59
.
24.
Spadoni
,
A.
, and
Daraio
,
C.
,
2010
, “
Generation and Control of Sound Bullets With a Nonlinear Acoustic Lens
,”
Proc. Natl. Acad. Sci.
,
107
(
16
), pp.
7230
7234
.
25.
Yang
,
J.
,
Silverstro
,
C.
,
Sangiorgio
,
S.
,
Borkowski
,
S.
,
Ebramzadeh
,
E.
,
Nardo
,
L.
, and
Daraio
,
C.
,
2012
, “
Nondestructive Evaluation of Orthopaedic Implant Stability in THA Using Highly Nonlinear Solitary Waves
,”
Smart Mater. Struct.
,
21
(
1
), p.
012002
.
26.
Donahue
,
C. M.
,
Anzel
,
P. W. J.
,
Bonanomi
,
L.
,
Keller
,
T.
, and
Daraio
,
C.
,
2014
, “
Experimental Realization of a Nonlinear Acoustic Lens With a Tunable Focus
,”
J. Appl. Phys.
,
104
, p.
014103
.
27.
Leonard
,
A.
,
Ponson
,
L.
, and
Daraio
,
C.
,
2014
, “
Exponential Stress Mitigation in Structured Granular Composites
,”
Extreme Mech. Lett.
,
1
, pp.
23
28
.
28.
Sievers
,
A.
, and
Takeno
,
S.
,
1988
, “
Intrinsic Localized Modes in Anharmonic Crystals
,”
Phys. Rev. Lett.
,
61
(
8
), pp.
970
973
.
29.
Campbell
,
D.
, and
Peyrard
,
M.
,
1990
,
CHAOS-Soviet American Perspectives on Nonlinear Science
,
American Institute of Physics
, New York.
30.
Theocharis
,
G.
,
Boechler
,
N.
,
Kevrekidis
,
P.
,
Job
,
S.
,
Porter
,
M.
, and
Daraio
,
C.
,
2010
, “
Intrinsic Energy Localization Through Discrete Gap Breathers in One-Dimensional Diatomic Granular Crystals
,”
Phys. Rev. E
,
82
(
5
), p.
056604
.
31.
Manevitch
,
L.
,
1999
,
Complex Representation of Dynamics of Coupled Nonlinear Oscillators: Mathematical Models of Non-Linear Excitations: Transfer, Dynamics, and Control in Condensed Systems and Other Media
,
Kluwer Academic
, New York.
32.
Kopidakis
,
G.
,
Aubry
,
S.
, and
Tsironis
,
G.
,
2001
, “
Targeted Energy Transfer Through Discrete Breathers in Nonlinear Systems
,”
Phys. Rev. Lett.
,
87
(
16
), p.
165501
.
33.
Maniadis
,
P.
, and
Aubry
,
S.
,
2005
, “
Targeted Energy Transfer by Fermi Resonance
,”
Physica D
,
202
(
3–4
), pp.
200
217
.
34.
Kosevich
,
Y.
,
Manevitch
,
L.
, and
Savin
,
A.
,
2009
, “
Energy Transfer in Weakly Coupled Nonlinear Oscillator Chains: Transition From a Wandering Breather to Nonlinear Self-Trapping
,”
J. Sound Vib.
,
322
(
3
), pp.
524
531
.
35.
Vorotnikov
,
K.
, and
Starosvetsky
,
Y.
,
2015a
, “
Nonlinear Energy Channeling in the Two-Dimensional, Locally Resonant, Unit-Cell Model. I. High Energy Pulsations and Routes to Energy Localization
,”
Chaos
,
25
(
7
), p.
073106
.
36.
Vorotnikov
,
K.
, and
Starosvetsky
,
Y.
,
2015b
, “
Nonlinear Energy Channeling in the Two-Dimensional, Locally Resonant, Unit-Cell Model. II. Low Energy Excitations and Unidirectional Energy Transport
,”
Chaos
,
25
(
7
), p.
073107
.
37.
Yao
,
C.
,
He.
,
Z.
, and
Zhan
,
M.
,
2013
, “
High-Frequency Forcing on Nonlinear Systems
,”
Chin. Phys. B
,
22
(
3
), p.
030503
.
38.
Bhadra
,
N.
,
Lahowetz
,
E. A.
,
Foldes
,
S. T.
, and
Kilgore
,
K. L.
,
2007
, “
Simulation of High-Frequency Sinusoidal Electrical Block of Mammalian Myelinated Axons
,”
J. Comput. Neurosci.
,
22
(
3
), pp.
313
326
.
39.
Koblischka
,
M. R.
,
Wei
,
J. D.
,
Kirsch
,
M.
, and
Hartmann
,
U.
,
2006
, “
High-Frequency Magnetic Force Microscopy-Imaging of Harddisk Write Heads
,”
Jpn. J. Appl. Phys.
,
45
, pp.
2238
2241
.
40.
Hunter
,
E. J.
,
Chant
,
R. J.
,
Wilkin
,
J. L.
, and
Kohut
,
J.
,
2010
, “
High-Frequency Forcing and Subtidal Response of the Hudson River Plume
,”
J. Geophys. Res.
,
115
(C7), p.
C07012
.
41.
Ylinen
,
A.
,
Bragin
,
A.
,
Nadasdy
,
Z.
,
Jando Sik
,
A.
, and
Buzsaki
,
G.
,
1995
, “
Sharp Wave-Associated High-Frequency Oscillation (200 Hz) in the Intact Hippocampus: Network and Intracellular Mechanisms
,”
J. Neurosci.
,
15
(
l
), pp.
30
46
.
42.
Levy
,
R.
,
Hutchison
,
W. D.
,
Lozano
,
A. M.
, and
Dostrovsky
,
J. O.
,
2000
, “
High-Frequency Synchronization of Neuronal Activity in the Subthalamic Nucleus of Parkinsonian Patients With Limb Tremor
,”
J. Neurosci.
,
20
(
20
), pp.
7766
7775
.
43.
Wang
,
Y.
, and
Lee
,
K.
,
1973
, “
Propagation of a Disturbance in a Chain of Interacting Harmonic Oscillators
,”
Am. J. Phys.
,
41
(
1
), pp.
51
54
.
44.
Zhang
,
Y.
,
Moore
,
K.
,
McFarland
,
D.
, and
Vakakis
,
A.
,
2015
, “
Targeted Energy Transfers and Passive Acoustic Wave Redirection in a Two-Dimensional Granular Network Under Periodic Excitation
,”
J. Appl. Phys.
,
118
(
23
), p.
234901
.
45.
Hasan
,
M.
,
Starosvetsky
,
Y.
,
Vakakis
,
A.
, and
Manevitch
,
L.
,
2013
, “
Nonlinear Targeted Energy Transfer and Macroscopic Analog of the Quantum Landau–Zener Effect in Coupled Granular Chains
,”
Physica D
,
252
, pp.
46
58
.
46.
Zener
,
C.
,
1932
, “
Non-Adiabatic Crossing of Energy Levels
,”
Proc. R. Soc. London A
,
137
(
833
), pp.
696
702
.
47.
Razavy
,
M.
,
2003
,
Quantum Theory of Tunneling
,
World Scientific
, Singapore.
You do not currently have access to this content.