In this study, the effect of energy dissipation on harmonic waves propagating in one-dimensional monoatomic linear viscoelastic mass-spring chains is investigated. In particular, first dispersion laws in terms of wavenumber, attenuation, and wave propagation velocities (phase, group, and energy) for a generic viscoelastic mass-spring chain are derived from the homologous linear elastic (LE) expressions in force of the correspondence principle. A new formula for the energy velocity is introduced to account for energy dissipation. Next, such relations are specified for the Kelvin Voigt (KV) and the standard linear solid (SLS) rheological models. The analysis of the KV mass-spring chain in the high-frequency regime proves that the so-called wavenumber-gap is not related to energy dissipation, as assumed in previous studies, but is due to the nonphysical rigid behavior of the model at high frequencies. The SLS mass-spring chain, in fact, does not show any wavenumber-gap and at high frequencies recovers the wavenumber dispersion curve of the LE system. The behavior of the energy velocity for the different mass-spring chains confirms this conclusion.

References

References
1.
Brillouin
,
L.
,
1946
,
Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices
,
Dover
,
New York
.
2.
Deymier
,
P. A.
, ed.,
2013
,
Acoustic Metamaterials and Phononic Crystals
, Vol. 173,
Springer
,
Berlin
.
3.
Mead
,
D.
,
1973
, “
A General Theory of Harmonic Wave Propagation in Linear Periodic Systems With Multiple Coupling
,”
J. Sound Vib.
,
27
(
2
), pp.
235
260
.
4.
Farzbod
,
F.
, and
Leamy
,
M. J.
,
2011
, “
Analysis of Bloch's Method in Structures With Energy Dissipation
,”
ASME J. Vib. Acoust.
,
133
(
5
), p.
051010
.
5.
Hussein
,
M. I.
,
2009
, “
Theory of Damped Bloch Waves in Elastic Media
,”
Phys. Rev. B
,
80
(
21
), p.
212301
.
6.
Hussein
,
M. I.
, and
Frazier
,
M. J.
,
2010
, “
Band Structure of Phononic Crystals With General Damping
,”
J. Appl. Phys.
,
108
(
9
), p.
093506
.
7.
Nesterenko
,
V. F.
,
2001
,
Dynamics of Heterogeneous Materials
,
Springer
,
New York
.
8.
Daraio
,
C.
,
Nesterenko
,
V.
,
Herbold
,
E.
, and
Jin
,
S.
,
2006
, “
Tunability of Solitary Wave Properties in One-Dimensional Strongly Nonlinear Phononic Crystals
,”
Phys. Rev. E
,
73
(
2
), p.
026610
.
9.
Narisetti
,
R. K.
,
Leamy
,
M. J.
, and
Ruzzene
,
M.
,
2010
, “
A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures
,”
ASME J. Vib. Acoust.
,
132
(
3
), p.
031001
.
10.
Smith
,
E.
, and
Ferri
,
A.
,
2015
, “
Shock Isolation in Finite-Length Dimer Chains With Linear, Cubic and Hertzian Spring Interactions
,”
ASME J. Vib. Acoust.
,
138
(
1
), p.
011012
.
11.
Newton
,
I.
,
1686
,
Principia-Book II
,
Imprimatur S. Pepys, Reg. Soc. Præses
,
London
.
12.
Kelvin
,
L.
,
1884
,
Popular Lectures
, Vol.
I
,
Macmillan
,
London
.
13.
Gao
,
Q.
,
Wu
,
F.
,
Zhang
,
H.
,
Zhong
,
W.
,
Howson
,
W.
, and
Williams
,
F.
,
2012
, “
Exact Solutions for Dynamic Response of a Periodic Spring and Mass Structure
,”
J. Sound Vib.
,
331
(
5
), pp.
1183
1190
.
14.
Wang
,
W.
,
Yu
,
J.
, and
Tang
,
Z.
,
2008
, “
General Dispersion and Dissipation Relations in a One-Dimensional Viscoelastic Lattice
,”
Phys. Lett. A
,
373
(
1
), pp.
5
8
.
15.
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
.
16.
Moiseyenko
,
R. P.
, and
Laude
,
V.
,
2011
, “
Material Loss Influence on the Complex Band Structure and Group Velocity in Phononic Crystals
,”
Phys. Rev. B
,
83
(
6
), p.
064301
.
17.
Christensen
,
R.
,
1982
,
Theory of Viscoelasticity: An Introduction
,
Academic Press
,
New York
.
18.
Bland
,
D.
,
1960
,
The Theory of Linear Viscoelasticity
(International Series of Monographs in Pure and Applied Mathematics),
Pergamon Press
,
Oxford, UK
.
19.
Carcione
,
J. M.
, ed.,
2007
, “
Viscoelasticity and Wave Propagation
,”
Handbook of Geophysical Exploration: Seismic Exploration
, Vol.
38
,
Pergamon
,
Amsterdam
.
20.
Mainardi
,
F.
, and
Spada
,
G.
,
2011
, “
Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology
,”
Eur. Phys. J.: Spec. Top.
,
193
(
1
), pp.
133
160
.
21.
Mainardi
,
F.
,
1987
, “
Energy Velocity for Hyperbolic Dispersive Waves
,”
Wave Motion
,
9
(
3
), pp.
201
208
.
22.
Mainardi
,
F.
, and
Van Groesen
,
E.
,
1989
, “
Energy Propagation in Linear Hyperbolic Systems
,”
Il Nuovo Cimento B Ser. 11
,
104
(
4
), pp.
487
496
.
You do not currently have access to this content.