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.
Issue Section:
Research Papers
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
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