Abstract

The present work aims to examine the metamaterial vibrational behavior of circular few-layer graphene sheets under layerwise tension forces. For this objective, a simplified three-membrane model is developed to simulate flexural vibration of tensioned circular few-layer graphene sheets, in which tensioned top and bottom layers are modeled as two elastic membranes while all less-tensioned or tension-free inner layers together are treated as a single membrane, and the three membranes are coupled through the van der Waals interaction between adjacent layers. Our results show that when the two outermost layers are highly tensioned but the inner layers are free of tension, circular few-layer graphene sheets exhibit negative effective mass within a certain terahertz frequency range. Moreover, such few-layer graphene sheets with negative effective mass demonstrate remarkable vibration isolation and vibration suppression. This research broadens our perspectives for designing and analyzing graphene-based metamaterials and resonators and could find potential application in nanoelectromechanical systems.

References

References
1.
Lee
,
S. H.
, and
Wright
,
O. B.
,
2016
, “
Origin of Negative Density and Modulus in Acoustic Metamaterials
,”
Phys. Rev. B
,
93
(
2
), p.
024302
. 10.1103/PhysRevB.93.024302
2.
Ma
,
G.
, and
Sheng
,
P.
,
2016
, “
Acoustic Metamaterials: From Local Resonances to Broad Horizons
,”
Sci. Adv.
,
2
(
2
), p.
e1501595
. 10.1126/sciadv.1501595
3.
Liu
,
Z. Y.
,
Zhang
,
X. X.
,
Mao
,
Y. W.
,
Zhu
,
Y. Y.
,
Yang
,
Z. Y.
,
Chan
,
C. T.
, and
Sheng
,
P.
,
2000
, “
Locally Resonant Sonic Materials
,”
Science
,
289
(
5485
), pp.
1734
1736
. 10.1126/science.289.5485.1734
4.
Milton
,
G. W.
, and
Willis
,
J. R.
,
2007
, “
On Modifications of Newton’s Second Law and Linear Continuum Elastodynamics
,”
Proc. R. Soc. A
,
463
(
2079
), pp.
855
880
. 10.1098/rspa.2006.1795
5.
Yao
,
S. S.
,
Zhou
,
X. M.
, and
Hu
,
G. K.
,
2008
, “
Experimental Study on Negative Effective Mass in a 1D Mass-Spring System
,”
New J. Phys.
,
10
(
4
), p.
043020
. 10.1088/1367-2630/10/4/043020
6.
Tian
,
Y. J.
,
Wu
,
J. H.
,
Li
,
H. L.
,
Gu
,
C. S.
,
Yang
,
Z. R.
,
Zhao
,
Z. T.
, and
Lu
,
K.
,
2019
, “
Elastic Wave Propagation in the Elastic Metamaterials Containing Parallel Multi-Resonators
,”
J. Phys. D: Appl. Phys.
,
52
(
39
), p.
395301
. 10.1088/1361-6463/ab2dba
7.
Dong
,
H. W.
,
Zhao
,
S. D.
,
Wang
,
Y. S.
, and
Zhang
,
C. Z.
,
2017
, “
Topology Optimization of Anisotropic Broadband Double-Negative Elastic Metamaterials
,”
J. Mech. Phys. Solids
,
105
, pp.
54
80
. 10.1016/j.jmps.2017.04.009
8.
Krushynska
,
A. O.
,
Miniaci
,
M.
,
Bosia
,
F.
, and
Pugno
,
N. M.
,
2017
, “
Coupling Local Resonance With Bragg Band Gaps in Single-Phase Mechanical Metamaterials
,”
Extreme Mech. Lett.
,
12
, pp.
30
36
. 10.1016/j.eml.2016.10.004
9.
Li
,
Z. W.
,
Wang
,
C.
, and
Wang
,
X. D.
,
2018
, “
Modelling of Elastic Metamaterials With Negative Mass and Modulus Based on Translational Resonance
,”
Int. J. Solids Struct.
,
162
, pp.
271
284
.
10.
Li
,
Z. W.
,
Hu
,
H.
, and
Wang
,
X. D.
,
2018
, “
A New Two-Dimensional Elastic Metamaterial System With Multiple Local Resonances
,”
Int. J. Mech. Sci.
,
149
, pp.
273
284
. 10.1016/j.ijmecsci.2018.09.053
11.
Liu
,
Y. Q.
,
Su
,
X. Y.
, and
Sun
,
C. T.
,
2015
, “
Broadband Elastic Metamaterial With Single Negativity by Mimicking Lattice Systems
,”
J. Mech. Phys. Solids
,
74
, pp.
158
174
. 10.1016/j.jmps.2014.09.011
12.
Xu
,
Y. C.
,
Wu
,
J. H.
,
Cai
,
Y. Q.
, and
Ma
,
F. Y.
,
2019
, “
Investigation on Dynamic Effective Parameters of Perforated Thin-Plate Acoustic Metamaterials
,”
J. Phys. D: Appl. Phys.
,
52
(
40
), p.
405301
. 10.1088/1361-6463/ab3011
13.
Sugino
,
C.
,
Ruzzene
,
M.
, and
Erturk
,
A.
,
2020
, “
An Analytical Framework for Locally Resonant Piezoelectric Metamaterial Plates
,”
Int. J. Solids Struct.
,
182–183
, pp.
281
294
. 10.1016/j.ijsolstr.2019.08.011
14.
Muhammad
, and
Lim
,
C. W.
,
2019
, “
Elastic Waves Propagation in Thin Plate Metamaterials and Evidence of Low Frequency Pseudo and Local Resonance Bandgaps
,”
Phys. Lett. A
,
383
(
23
), pp.
2789
2796
. 10.1016/j.physleta.2019.05.039
15.
Chen
,
Y. Y.
,
Hu
,
G. K.
, and
Huang
,
G. L.
,
2017
, “
A Hybrid Elastic Metamaterial With Negative Mass Density and Tunable Bending Stiffness
,”
J. Mech. Phys. Solids
,
105
, pp.
179
198
. 10.1016/j.jmps.2017.05.009
16.
Wang
,
W.
,
Bonello
,
B.
,
Djafari-Rouhani
,
B.
,
Pennec
,
Y.
, and
Zhao
,
J. F.
,
2018
, “
Double-Negative Pillared Elastic Metamaterial
,”
Phys. Rev. Appl.
,
10
(
6
), p.
064011
. 10.1103/PhysRevApplied.10.064011
17.
Chuang
,
K. C.
,
Lv
,
X. F.
, and
Wang
,
Y. H.
,
2019
, “
A Bandgap Switchable Elastic Metamaterial Using Shape Memory Alloys
,”
J. Appl. Phys.
,
125
(
5
), p.
055101
. 10.1063/1.5065557
18.
Sang
,
S.
,
Mhannawee
,
A.
, and
Wang
,
Z. P.
,
2019
, “
A Design of Active Elastic Metamaterials With Negative Mass Density and Tunable Bulk Modulus
,”
Acta Mech.
,
230
(
3
), pp.
1003
1008
. 10.1007/s00707-018-2320-2
19.
Novoselov
,
K. S.
,
Geim
,
A. K.
,
Morozov
,
S. V.
,
Jiang
,
D.
,
Zhang
,
Y.
,
Dubonos
,
S. V.
,
Grigorieva
,
I. V.
, and
Firsov
,
A. A.
,
2004
, “
Electric Field Effect in Atomically Thin Carbon Films
,”
Science
,
306
(
5696
), pp.
666
669
. 10.1126/science.1102896
20.
Lee
,
C.
,
Wei
,
X.
,
Kysar
,
J. W.
, and
Hone
,
J.
,
2008
, “
Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene
,”
Science
,
321
(
5887
), pp.
385
388
. 10.1126/science.1157996
21.
Castro Neto
,
A. H.
,
Guinea
,
F.
,
Peres
,
N. M. R.
,
Novoselov
,
K. S.
, and
Geim
,
A. K.
,
2009
, “
The Electronic Properties of Graphene
,”
Rev. Mod. Phys.
,
81
(
1
), pp.
109
162
. 10.1103/RevModPhys.81.109
22.
Cao
,
Y.
,
Fatemi
,
V.
,
Fang
,
S.
,
Watanabe
,
K.
,
Taniguchi
,
T.
,
Kaxiras
,
E.
, and
Jarillo-Herrero
,
P.
,
2018
, “
Unconventional Superconductivity in Magic-Angle Graphene Superlattices
,”
Nature
,
556
(
7699
), pp.
43
50
. 10.1038/nature26160
23.
Bunch
,
J. S.
,
van der Zande
,
A. M.
,
Verbridge
,
S. S.
,
Frank
,
I. W.
,
Tanenbaum
,
D. M.
,
Parpia
,
J. M.
,
Craighead
,
H. G.
, and
McEuen
,
P. L.
,
2007
, “
Electromechanical Resonators From Graphene Sheets
,”
Science
,
315
(
5811
), pp.
490
492
. 10.1126/science.1136836
24.
Garcia-Sanchez
,
D.
,
van der Zande
,
A. M.
,
Paulo
,
A. S.
,
Lassagne
,
B.
,
McEuen
,
P. L.
, and
Bachtold
,
A.
,
2008
, “
Imaging Mechanical Vibrations in Suspended Graphene Sheets
,”
Nano Lett.
,
8
(
5
), pp.
1399
1403
. 10.1021/nl080201h
25.
Castellanos-Gomez
,
A.
,
Singh
,
V.
,
van der Zant
,
H. S. J.
, and
Steele
,
G. A.
,
2015
, “
Mechanics of Freely-Suspended Ultrathin Layered Materials
,”
Ann. Phys.
,
527
(
1–2
), pp.
27
44
. 10.1002/andp.201400153
26.
Khan
,
Z. H.
,
Kermany
,
A. R.
,
Öchsner
,
A.
, and
Iacopi
,
F.
,
2017
, “
Mechanical and Electromechanical Properties of Graphene and Their Potential Application in MEMS
,”
J. Phys. D: Appl. Phys.
,
50
(
5
), p.
053003
. 10.1088/1361-6463/50/5/053003
27.
Robinson
,
J. T.
,
Zalalutdinov
,
M.
,
Baldwin
,
J. W.
,
Snow
,
E. S.
,
Wei
,
Z. Q.
,
Sheehan
,
P.
, and
Houston
,
B. H.
,
2008
, “
Wafer-Scale Reduced Graphene Oxide Films for Nanomechanical Devices
,”
Nano Lett.
,
8
(
10
), pp.
3441
3445
. 10.1021/nl8023092
28.
Weber
,
P.
,
Güttinger
,
J.
,
Tsioutsios
,
I.
,
Chang
,
D. E.
, and
Bachtold
,
A.
,
2014
, “
Coupling Graphene Mechanical Resonators to Superconducting Microwave Cavities
,”
Nano Lett.
,
14
(
5
), pp.
2854
2860
. 10.1021/nl500879k
29.
Zalalutdinov
,
M. K.
,
Robinson
,
J. T.
,
Junkermeier
,
C. E.
,
Culbertson
,
J. C.
,
Reinecke
,
T. L.
,
Stine
,
R.
,
Sheehan
,
P. E.
,
Houston
,
B. H.
, and
Snow
,
E. S.
,
2012
, “
Engineering Graphene Mechanical Systems
,”
Nano Lett.
,
12
(
8
), pp.
4212
4218
. 10.1021/nl3018059
30.
Poot
,
M.
, and
van der Zant
,
H. S. J.
,
2008
, “
Nanomechanical Properties of Few-Layer Graphene Membranes
,”
Appl. Phys. Lett.
,
92
(
6
), p.
063111
. 10.1063/1.2857472
31.
Wong
,
C. L.
,
Annamalai
,
M.
,
Wang
,
Z. Q.
, and
Palaniapan
,
M.
,
2010
, “
Characterization of Nanomechanical Graphene Drum Structures
,”
J. Micromech. Microeng.
,
20
(
11
), p.
115029
. 10.1088/0960-1317/20/11/115029
32.
Singh
,
V.
,
Bosman
,
S. J.
,
Schneider
,
B. H.
,
Blanter
,
Y. M.
,
Castellanos-Gomez
,
A.
, and
Steele
,
G. A.
,
2014
, “
Optomechanical Coupling Between a Multilayer Graphene Mechanical Resonator and a Superconducting Microwave Cavity
,”
Nat. Nanotech.
,
9
(
10
), pp.
820
824
. 10.1038/nnano.2014.168
33.
Lu
,
L.
,
Ru
,
C. Q.
, and
Guo
,
X. M.
,
2020
, “
Vibration Isolation of Few-Layer Graphene Sheets
,”
Int. J. Solids Struct.
,
185–186
, pp.
78
88
. 10.1016/j.ijsolstr.2019.08.029
34.
Noga
,
S.
,
2010
, “
Free Transverse Vibration Analysis of an Elastically Connected Annular and Circular Double-Membrane Compound System
,”
J. Sound Vib.
,
329
(
9
), pp.
1507
1522
. 10.1016/j.jsv.2009.10.039
35.
Kang
,
J. H.
,
2017
, “
Viscously Damped Free and Forced Vibrations of Circular and Annular Membranes by a Closed Form Exact Method
,”
Thin-Walled Struct.
,
116
, pp.
194
200
. 10.1016/j.tws.2017.03.027
36.
Dienwiebel
,
M.
,
Verhoeven
,
G. S.
,
Pradeep
,
N.
,
Frenken
,
J. W. M.
,
Heimberg
,
J. A.
, and
Zandbergen
,
H. W.
,
2004
, “
Superlubricity of Graphite
,”
Phys. Rev. Lett.
,
92
(
12
), p.
126101
. 10.1103/PhysRevLett.92.126101
37.
Xu
,
L.
,
Ma
,
T.
,
Hu
,
Y.
, and
Wang
,
H.
,
2012
, “
Molecular Dynamics Simulation of the Interlayer Sliding Behavior in Few-Layer Graphene
,”
Carbon
,
50
(
3
), pp.
1025
1032
. 10.1016/j.carbon.2011.10.006
38.
Koshino
,
M.
,
2015
, “
Interlayer Interaction in General Incommensurate Atomic Layers
,”
New J. Phys.
,
17
(
1
), p.
015014
. 10.1088/1367-2630/17/1/015014
39.
Han
,
J.
,
Ryu
,
S.
,
Kim
,
D. K.
,
Woo
,
W.
, and
Sohn
,
D.
,
2016
, “
Effect of Interlayer Sliding on the Estimation of Elastic Modulus of Multilayer Graphene in Nanoindentation Simulation
,”
EPL
,
114
(
6
), p.
68001
. 10.1209/0295-5075/114/68001
40.
Lu
,
Q.
,
Arroyo
,
M.
, and
Huang
,
R.
,
2009
, “
Elastic Bending Modulus of Monolayer Graphene
,”
J. Phys. D: Appl. Phys.
,
42
(
10
), p.
102002
. 10.1088/0022-3727/42/10/102002
41.
Wei
,
Y.
,
Wang
,
B.
,
Wu
,
J.
,
Yang
,
R.
, and
Dunn
,
M. L.
,
2013
, “
Bending Rigidity and Gaussian Bending Stiffness of Single-Layered Graphene
,”
Nano Lett.
,
13
(
1
), pp.
26
30
. 10.1021/nl303168w
42.
Barton
,
R. A.
,
Ilic
,
B.
,
van der Zande
,
A. M.
,
Whitney
,
W. S.
,
McEuen
,
P. L.
,
Parpia
,
J. M.
, and
Craighead
,
H. G.
,
2011
, “
High, Size-Dependent Quality Factor in an Array of Graphene Mechanical Resonators
,”
Nano Lett.
,
11
(
3
), pp.
1232
1236
. 10.1021/nl1042227
43.
Chen
,
C. Y.
,
Lee
,
S. W.
,
Deshpande
,
V. V.
,
Lee
,
G. H.
,
Lekas
,
M.
,
Shepard
,
K.
, and
Hone
,
J.
,
2013
, “
Graphene Mechanical Oscillators With Tunable Frequency
,”
Nat. Nanotech.
,
8
(
12
), pp.
923
927
. 10.1038/nnano.2013.232
44.
Wang
,
C. Y.
,
Ru
,
C. Q.
, and
Mioduchowski
,
A.
,
2005
, “
Axisymmetric and Beamlike Vibrations of Multiwall Carbon Nanotubes
,”
Phys. Rev. B
,
72
(
7
), p.
075414
. 10.1103/PhysRevB.72.075414
45.
Lu
,
L.
,
Ru
,
C. Q.
, and
Guo
,
X. M.
,
2017
, “
Vibration of a Multilayer Graphene Sheet Under Layerwise Tension Forces
,”
Int. J. Mech. Sci.
,
121
, pp.
157
163
. 10.1016/j.ijmecsci.2017.01.007
46.
Rasool
,
H. I.
,
Ophus
,
C.
,
Klug
,
W. S.
,
Zettl
,
A.
, and
Gimzewski
,
J. K.
,
2013
, “
Measurement of the Intrinsic Strength of Crystalline and Polycrystalline Graphene
,”
Nat. Commun.
,
4
(
1
), p.
2811
. 10.1038/ncomms3811
47.
Wang
,
L.
,
Chen
,
X.
,
Yu
,
A.
,
Zhang
,
Y.
,
Ding
,
J.
, and
Lu
,
W.
,
2014
, “
Highly Sensitive and Wide-Band Tunable Terahertz Response of Plasma Waves Based on Graphene Field Effect Transistors
,”
Sci. Rep.
,
4
(
1
), p.
5470
. 10.1038/srep05470
48.
Jnawali
,
G.
,
Rao
,
Y.
,
Yan
,
H.
, and
Heinz
,
T. F.
,
2013
, “
Observation of a Transient Decrease in Terahertz Conductivity of Single-Layer Graphene Induced by Ultrafast Optical Excitation
,”
Nano Lett.
,
13
(
2
), pp.
524
530
. 10.1021/nl303988q
49.
Ren
,
L.
,
Zhang
,
Q.
,
Yao
,
J.
,
Sun
,
Z.
,
Kaneko
,
R.
,
Yan
,
Z.
,
Nanot
,
S.
,
Jin
,
Z.
,
Kawayama
,
I.
,
Tonouchi
,
M.
,
Tour
,
J. M.
, and
Kono
,
J.
,
2012
, “
Terahertz and Infrared Spectroscopy of Gated Large-Area Graphene
,”
Nano Lett.
,
12
(
7
), pp.
3711
3715
. 10.1021/nl301496r
50.
Svintsov
,
D.
,
Leiman
,
V. G.
,
Ryzhii
,
V.
,
Otsuji
,
T.
, and
Shur
,
M. S.
,
2014
, “
Graphene Nanoelectromechanical Resonators for the Detection of Modulated Terahertz Radiation
,”
J. Phys. D: Appl. Phys.
,
47
(
50
), p.
505105
. 10.1088/0022-3727/47/50/505105
51.
Duan
,
W. H.
, and
Wang
,
C. M.
,
2009
, “
Nonlinear Bending and Stretching of a Circular Graphene Sheet Under a Central Point Load
,”
Nanotechnology
,
20
(
7
), p.
075702
. 10.1088/0957-4484/20/7/075702
52.
Komaragiri
,
U.
,
Begley
,
M. R.
, and
Simmonds
,
J. G.
,
2005
, “
The Mechanical Response of Freestanding Circular Elastic Films Under Point and Pressure Loads
,”
ASME J. Appl. Mech.
,
72
(
2
), pp.
203
212
. 10.1115/1.1827246
53.
Yang
,
B.
,
Chen
,
W. Q.
, and
Ding
,
H. J.
,
2014
, “
Approximate Elasticity Solutions for Functionally Graded Circular Plates Subject to a Concentrated Force at the Center
,”
Math. Mech. Solids
,
19
(
3
), pp.
277
288
. 10.1177/1081286512463572
You do not currently have access to this content.