This manuscript presents a new nonlocal homogenization model (NHM) for wave dispersion and attenuation in elastic and viscoelastic periodic layered media. Homogenization with multiple spatial scales based on asymptotic expansions of up to eighth order is employed to formulate the proposed nonlocal homogenization model. A momentum balance equation, nonlocal in both space and time, is formulated consistent with the gradient elasticity theory. A key contribution in this regard is that all model coefficients including high-order length-scale parameters are derived directly from microstructural material properties and geometry. The capability of the proposed model in capturing the characteristics of wave propagation in heterogeneous media is demonstrated in multiphase elastic and viscoelastic materials. The nonlocal homogenization model is shown to accurately predict wave dispersion and attenuation within the acoustic regime for both elastic and viscoelastic layered composites.

References

References
1.
Schurig
,
D.
,
Mock
,
J. J.
,
Justice
,
B. J.
,
Cummer
,
S. A.
,
Pendry
,
J. B.
,
Starr
,
A. F.
, and
Smith
,
D. R.
,
2006
, “
Metamaterial Electromagnetic Cloak at Microwave Frequencies
,”
Science
,
314
(
5801
), pp.
977
980
.
2.
Zhang
,
S.
,
Xia
,
C.
, and
Fang
,
N.
,
2011
, “
Broadband Acoustic Cloak for Ultrasound Waves
,”
Phys. Rev. Lett.
,
106
(
2
), p.
024301
.
3.
Gonella
,
S.
,
To
,
A. C.
, and
Liu
,
W. K.
,
2009
, “
Interplay Between Phononic Bandgaps and Piezoelectric Microstructures for Energy Harvesting
,”
J. Mech. Phys. Solids
,
57
(
3
), pp.
621
633
.
4.
Carrara
,
M.
,
Cacan
,
M. R.
,
Toussaint
,
J.
,
Leamy
,
M. J.
,
Ruzzene
,
M.
, and
Erturk
,
A.
,
2013
, “
Metamaterial-Inspired Structures and Concepts for Elastoacoustic Wave Energy Harvesting
,”
Smart Mater. Struct.
,
22
(
6
), p.
065004
.
5.
Naify
,
C. J.
,
Chang
,
C. M.
,
McKnight
,
G.
, and
Nutt
,
S.
,
2011
, “
Transmission Loss of Membrane-Type Acoustic Metamaterials With Coaxial Ring Masses
,”
J. Appl. Phys.
,
110
(
12
), p.
124903
.
6.
Tan
,
K. T.
,
Huang
,
H. H.
, and
Sun
,
C. T.
,
2014
, “
Blast-Wave Impact Mitigation Using Negative Effective Mass Density Concept of Elastic Metamaterials
,”
Int. J. Impact Eng.
,
64
, pp.
20
29
.
7.
Pennec
,
Y.
,
Vasseur
,
J. O.
,
Djafari-Rouhani
,
B.
,
Dobrzyński
,
L.
, and
Deymier
,
P. A.
,
2010
, “
Two-Dimensional Phononic Crystals: Examples and Applications
,”
Surf. Sci. Rep.
,
65
(
8
), pp.
229
291
.
8.
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
.
9.
Liu
,
Z.
,
Zhang
,
X.
,
Mao
,
Y.
,
Zhu
,
Y. Y.
,
Yang
,
Z.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2000
, “
Locally Resonant Sonic Materials
,”
Science
,
289
(
5485
), pp.
1734
1736
.
10.
Ma
,
G.
, and
Sheng
,
P.
,
2016
, “
Acoustic Metamaterials: From Local Resonances to Broad Horizons
,”
Sci. Adv.
,
2
(
2
), p.
e1501595
.
11.
Mindlin
,
R. D.
,
1964
, “
Micro-Structure in Linear Elasticity
,”
Arch. Ration. Mech. Anal.
,
16
(
1
), pp.
51
78
.
12.
Achenbach
,
J. D.
, and
Herrmann
,
G.
,
1968
, “
Dispersion of Free Harmonic Waves in Fiber-Reinforced Composites
,”
AIAA J.
,
6
(
10
), pp.
1832
1836
.
13.
Hussein
,
M. I.
,
Hulbert
,
G. M.
, and
Scott
,
R. A.
,
2006
, “
Dispersive Elastodynamics of 1D Banded Materials and Structures: Analysis
,”
J. Sound Vib.
,
289
(
4–5
), pp.
779
806
.
14.
Sigalas
,
M. M.
, and
Economou
,
E. N.
,
1992
, “
Elastic and Acoustic Wave Band Structure
,”
J. Sound Vib.
,
158
(
2
), pp.
377
382
.
15.
Mitchell
,
S. J.
,
Pandolfi
,
A.
, and
Ortiz
,
M.
,
2015
, “
Investigation of Elastic Wave Transmission in a Metaconcrete Slab
,”
Mech. Mater.
,
91
(Part 1), pp.
295
303
.
16.
Bensoussan
,
A.
,
Lions
,
J.-L.
, and
Papanicolaou
,
G.
,
1978
,
Asymptotic Analysis for Periodic Structures
,
North-Holland
,
Amsterdam, Netherlands
.
17.
Sanchez-Palencia
,
E.
,
1980
, “
Non-Homogeneous Media and Vibration Theory
,”
Lecture Notes in Physics
, Vol.
127
, Springer, Berlin, Germany.
18.
Boutin
,
C.
, and
Auriault
,
J. L.
,
1993
, “
Rayleigh Scattering in Elastic Composite Materials
,”
Int. J. Eng. Sci.
,
31
(
12
), pp.
1669
1689
.
19.
Fish
,
J.
, and
Chen
,
W.
,
2001
, “
Higher-Order Homogenization of Initial/Boundary-Value Problem
,”
J. Eng. Mech.
,
127
(
12
), pp.
1223
1230
.
20.
Fish
,
J.
,
Chen
,
W.
, and
Nagai
,
G.
,
2002
, “
Non-Local Dispersive Model for Wave Propagation in Heterogeneous Media: One-Dimensional Case
,”
Int. J. Numer. Methods Eng.
,
54
(
3
), pp.
331
346
.
21.
Fish
,
J.
,
Chen
,
W.
, and
Nagai
,
G.
,
2002
, “
Non-Local Dispersive Model for Wave Propagation in Heterogeneous Media: Multi-Dimensional Case
,”
Int. J. Numer. Methods Eng.
,
54
(
3
), pp.
347
363
.
22.
Andrianov
,
I. V.
,
Bolshakov
,
V. I.
,
Danishevs'kyy
,
V. V.
, and
Weichert
,
D.
,
2008
, “
Higher Order Asymptotic Homogenization and Wave Propagation in Periodic Composite Materials
,”
Proc. R. Soc., London, Ser. A
,
464
(
2093
), pp.
1181
1201
.
23.
Hui
,
T.
, and
Oskay
,
C.
,
2014
, “
A High Order Homogenization Model for Transient Dynamics of Heterogeneous Media Including Micro-Inertia Effects
,”
Comput. Methods Appl. Mech. Eng.
,
273
, pp.
181
203
.
24.
Auriault
,
J. L.
, and
Boutin
,
C.
,
2012
, “
Long Wavelength Inner-Resonance Cut-Off Frequencies in Elastic Composite Materials
,”
Int. J. Solids Struct.
,
49
(
23–24
), pp.
3269
3281
.
25.
Craster
,
R. V.
,
Kaplunov
,
J.
, and
Pichugin
,
A. V.
,
2010
, “
High-Frequency Homogenization for Periodic Media
,”
Proc. R. Soc. London, Ser. A
,
466
(
2120
), pp.
2341
2362
.
26.
Nemat-Nasser
,
S.
,
Willis
,
J. R.
,
Srivastava
,
A.
, and
Amirkhizi
,
A. V.
,
2011
, “
Homogenization of Periodic Elastic Composites and Locally Resonant Sonic Materials
,”
Phys. Rev. B: Condens. Matter Mater. Phys.
,
83
(
10
), p.
104103
.
27.
Nemat-Nasser
,
S.
, and
Srivastava
,
A.
,
2011
, “
Overall Dynamic Constitutive Relations of Layered Elastic Composites
,”
J. Mech. Phys. Solids
,
59
(
10
), pp.
1953
1965
.
28.
Nassar
,
H.
,
He
,
Q.-C.
, and
Auffray
,
N.
,
2015
, “
Willis Elastodynamic Homogenization Theory Revisited for Periodic Media
,”
J. Mech. Phys. Solids
,
77
, pp.
158
178
.
29.
Nassar
,
H.
,
He
,
Q.-C.
, and
Auffray
,
N.
,
2016
, “
On Asymptotic Elastodynamic Homogenization Approaches for Periodic Media
,”
J. Mech. Phys. Solids
,
88
, pp.
274
290
.
30.
Pham
,
K.
,
Kouznetsova
,
V. G.
, and
Geers
,
M. G. D.
,
2013
, “
Transient Computational Homogenization for Heterogeneous Materials Under Dynamic Excitation
,”
J. Mech. Phys. Solids
,
61
(
11
), pp.
2125
2146
.
31.
Sridhar
,
A.
,
Kouznetsova
,
V. G.
, and
Geers
,
M. G. D.
,
2016
, “
Homogenization of Locally Resonant Acoustic Metamaterials Towards an Emergent Enriched Continuum
,”
Comput. Mech.
,
57
(
3
), pp.
423
435
.
32.
Filonova
,
V.
,
Fafalis
,
D.
, and
Fish
,
J.
,
2016
, “
Dispersive Computational Continua
,”
Comput. Methods Appl. Mech. Eng.
,
298
, pp.
58
79
.
33.
Fafalis
,
D.
, and
Fish
,
J.
,
2015
, “
Computational Aspects of Dispersive Computational Continua for Elastic Heterogeneous Media
,”
Comput. Mech.
,
56
(
6
), pp.
931
946
.
34.
Askes
,
H.
,
Metrikine
,
A. V.
,
Pichugin
,
A. V.
, and
Bennett
,
T.
,
2008
, “
Four Simplified Gradient Elasticity Models for the Simulation of Dispersive Wave Propagation
,”
Philos. Mag.
,
88
(
28–29
), pp.
3415
3443
.
35.
Metrikine
,
A. V.
,
2006
, “
On Causality of the Gradient Elasticity Models
,”
J. Sound Vib.
,
297
(
3–5
), pp.
727
742
.
36.
Dontsov
,
E. V.
,
Tokmashev
,
R. D.
, and
Guzina
,
B. B.
,
2013
, “
A Physical Perspective of the Length Scales in Gradient Elasticity Through the Prism of Wave Dispersion
,”
Int. J. Solids Struct.
,
50
(
22–23
), pp.
3674
3684
.
37.
Askes
,
H.
, and
Aifantis
,
E. C.
,
2011
, “
Gradient Elasticity in Statics and Dynamics: An Overview of Formulations, Length Scale Identification Procedures, Finite Element Implementations and New Results
,”
Int. J. Solids Struct.
,
48
(
13
), pp.
1962
1990
.
38.
Liu
,
Y.
,
Yu
,
D.
,
Zhao
,
H.
,
Wen
,
J.
, and
Wen
,
X.
,
2008
, “
Theoretical Study of Two-Dimensional Phononic Crystals With Viscoelasticity Based on Fractional Derivative Models
,”
J. Phys. D: Appl. Phys.
,
41
(
6
), p.
065503
.
39.
Merheb
,
B.
,
Deymier
,
P. A.
,
Jain
,
M.
,
Aloshyna-Lesuffleur
,
M.
,
Mohanty
,
S.
,
Berker
,
A.
, and
Greger
,
R. W.
,
2008
, “
Elastic and Viscoelastic Effects in Rubber/Air Acoustic Band Gap Structures: A Theoretical and Experimental Study
,”
J. Appl. Phys.
,
104
(
6
), p.
064913
.
40.
Merheb
,
B.
,
Deymier
,
P. A.
,
Muralidharan
,
K.
,
Bucay
,
J.
,
Jain
,
M.
,
Aloshyna-Lesuffleur
,
M.
,
Greger
,
R. W.
,
Mohanty
,
S.
, and
Berker
,
A.
,
2009
, “
Viscoelastic Effect on Acoustic Band Gaps in Polymer-Fluid Composites
,”
Modell. Simul. Mater. Sci. Eng.
,
17
(
7
), p.
075013
.
41.
Hussein
,
M. I.
, and
Frazier
,
M. J.
,
2010
, “
Band Structure of Phononic Crystals With General Damping
,”
J. Appl. Phys.
,
108
(
9
), p.
093506
.
42.
Zhao
,
Y. P.
, and
Wei
,
P. J.
,
2009
, “
The Band Gap of 1D Viscoelastic Phononic Crystal
,”
Comput. Mater. Sci.
,
46
(
3
), pp.
603
606
.
43.
Krushynska
,
A. O.
,
Kouznetsova
,
V. G.
, and
Geers
,
M. G. D.
,
2016
, “
Visco-Elastic Effects on Wave Dispersion in Three-Phase Acoustic Metamaterials
,”
J. Mech. Phys. Solids
,
96
, pp.
29
47
.
44.
Hui
,
T.
, and
Oskay
,
C.
,
2013
, “
A Nonlocal Homogenization Model for Wave Dispersion in Dissipative Composite Materials
,”
Int. J. Solids Struct.
,
50
(
1
), pp.
38
48
.
45.
Hui
,
T.
, and
Oskay
,
C.
,
2015
, “
Laplace-Domain, High-Order Homogenization for Transient Dynamic Response of Viscoelastic Composites
,”
Int. J. Numer. Methods Eng.
,
103
(
13
), pp.
937
957
.
46.
Gambin
,
B.
, and
Kröner
,
E.
,
1989
, “
Higher-Order Terms in the Homogenized Stress-Strain Relation of Periodic Elastic Media
,”
Phys. Status Solidi B
,
151
(
2
), pp.
513
519
.
47.
Smyshlyaev
,
V. P.
, and
Cherednichenko
,
K. D.
,
2000
, “
On Rigorous Derivation of Strain Gradient Effects in the Overall Behaviour of Periodic Heterogeneous Media
,”
J. Mech. Phys. Solids
,
48
(
6
), pp.
1325
1357
.
48.
Auriault
,
J. L.
,
Geindreau
,
C.
, and
Boutin
,
C.
,
2005
, “
Filtration law in Porous Media With Poor Separation of Scales
,”
Transp. Porous Media
,
60
(
1
), pp.
89
108
.
49.
El Boudouti
,
E. H.
,
Djafari-Rouhani
,
B.
,
Akjouj
,
A.
, and
Dobrzynski
,
L.
,
2009
, “
Acoustic Waves in Solid and Fluid Layered Materials
,”
Surf. Sci. Rep.
,
64
(
11
), pp.
471
594
.
50.
Hu
,
R.
,
Prakash
,
C.
,
Tomar
,
V.
,
Harr
,
M.
,
Gunduz
,
I. E.
, and
Oskay
,
C.
,
2016
, “
Experimentally-Validated Mesoscale Modeling of the Coupled Mechanical–Thermal Response of AP–HTPB Energetic Material Under Dynamic Loading
,”
Int. J. Fract.
, pp.
1
22
, (epub).
51.
Oskay
,
C.
, and
Fish
,
J.
,
2007
, “
Eigendeformation-Based Reduced Order Homogenization for Failure Analysis of Heterogeneous Materials
,”
Comput. Methods Appl. Mech. Eng.
,
196
(
7
), pp.
1216
1243
.
52.
Oskay
,
C.
, and
Pal
,
G.
,
2010
, “
A Multiscale Failure Model for Analysis of Thin Heterogeneous Plates
,”
Int. J. Damage Mech.
,
19
(
5
), pp.
575
610
.
53.
Pichugin
,
A. V.
,
Askes
,
H.
, and
Tyas
,
A.
,
2008
, “
Asymptotic Equivalence of Homogenisation Procedures and Fine-Tuning of Continuum Theories
,”
J. Sound Vib.
,
313
(
3
), pp.
858
874
.
54.
Chen
,
W.
, and
Fish
,
J.
,
2001
, “
A Dispersive Model for Wave Propagation in Periodic Heterogeneous Media Based on Homogenization With Multiple Spatial and Temporal Scales
,”
ASME J. Appl. Mech.
,
68
(
2
), pp.
153
161
.
55.
Andrianov
,
I. V.
,
Awrejcewicz
,
J.
, and
Weichert
,
D.
,
2009
, “
Improved Continuous Models for Discrete Media
,”
Math. Probl. Eng.
,
2010
, p.
152
.
56.
Kaplunov
,
J. D.
, and
Pichugin
,
A. V.
,
2009
, “
On Rational Boundary Conditions for Higher-Order Long-Wave Models
,”
IUTAM Symposium on Scaling in Solid Mechanics
, Borodich, F.M., ed.,
Springer
,
Berlin, Germany
, pp.
81
90
.
57.
Polizzotto
,
C.
,
2003
, “
Gradient Elasticity and Nonstandard Boundary Conditions
,”
Int. J. Solids Struct.
,
40
(
26
), pp.
7399
7423
.
58.
Bedford
,
A.
, and
Drumheller
,
D. S.
,
1994
,
Elastic Wave Propagation
,
Wiley
, New York.
59.
Brillouin
,
L.
,
1953
,
Wave Propagation in Periodic Structures
,
2nd ed.
,
Dover Publication
, New York.
60.
Boutin
,
C.
,
1996
, “
Microstructural Effects in Elastic Composites
,”
Int. J. Solids Struct.
,
33
(
7
), pp.
1023
1051
.
61.
Brancik
,
L.
,
1999
, “
Programs for Fast Numerical Inversion of Laplace Transforms in Matlab Language Environment
,”
Konference MATLAB
‘99-Praha, pp.
27
39
.
You do not currently have access to this content.