This paper is concerned with the derivation of the exact solutions for the static responses of the simply supported flexoelectric nanobeams subjected to the applied mechanical load and applied voltage difference across the thickness of the beams. Considering both the direct and the converse flexoelectric effects, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements and the electric potential in the beams. Due to the converse flexoelectric effect, the active beams significantly counteract the applied mechanical load. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect in the beams resulting in the coupling of bending and stretching deformations in the beams. For the particular values of the length of the beam and the applied voltage, the deflection of the nanobeam due to the converse flexoelectric effect significantly increases with the decrease in the thickness of the beam. But the deflection of the beam remains invariant with the change in length of the beam for the particular values of the thickness of the beam and the applied voltage. Also, for the particular values of the thickness of the beam and the applied mechanical load, the induced transverse electric polarization on the surface of the beam is independent of the variation of the length of the beam and the value of the polarization increases with the decrease in the thickness of the beam. The benchmark results presented here may be useful for verifying further research and the present study suggests that the flexoelectric nanobeams may be effectively exploited for advanced applications as smart sensors and actuators at nanoscale.

References

References
1.
Mashkevich
,
V. S.
, and
Tolpygo
,
K. B.
,
1957
, “
Electrical, Optical and Elastic Properties of Diamond Type Crystals. I
,”
Sov. Phys.-JETP
,
5
(
3
), pp.
435
439
.
2.
Kogan
,
S. M.
,
1964
, “
Piezoelectric Effect During Inhomogeneous Deformation and Acoustic Scattering of Carriers in Crystals
,”
Sov. Phys.-Solid State
,
5
(
10
), pp.
2069
2070
.
3.
Harris
,
P.
,
1965
, “
Mechanism for the Shock Polarization of Dielectrics
,”
J. Appl. Phys.
,
36
(3), pp.
739
741
.10.1063/1.1714210
4.
Mindlin
,
R. D.
,
1968
, “
Polarization Gradient in Elastic Dielectrics
,”
Int. J. Solids Struct.
,
4
(6), pp.
637
642
.10.1016/0020-7683(68)90079-6
5.
Askar
,
A.
,
Lee
,
P. C. Y.
, and
Cakmak
,
A. S.
,
1970
, “
Lattice-Dynamics Approach to the Theory of Elastic Dielectrics With Polarization Gradient
,”
Phys. Rev. B
,
1
(8), pp.
3525
3530
.10.1103/PhysRevB.1.3525
6.
Bursian
,
E. V.
, and
Trunov
,
N. N.
,
1974
, “
Nonlocal Piezoelectric Effect
,”
Sov. Phys.-Solid State
,
16
(4), pp.
760
762
.
7.
Meyer
,
R. B.
,
1969
, “
Piezoelectric Effects in Liquid Crystals
,”
Phys. Rev. Lett.
,
22
(18), pp.
918
921
.10.1103/PhysRevLett.22.918
8.
Indenbom
,
V. L.
,
Loginov
,
E. B.
, and
Osipov
,
M. A.
,
1981
, “
Flexoelectric Effect and the Structure of Crystals
,”
Kristalografija
,
26
, pp.
1157
1162
.
9.
Tagantsev
,
A. K.
,
1986
, “
Piezoelectricity and Flexoelectricity in Crystalline Dielectrics
,”
Phys. Rev. B
,
34
(8), pp.
5883
5889
.10.1103/PhysRevB.34.5883
10.
Ma
,
W.
, and
Eric Cross
,
L.
,
2001
, “
Large Flexoelectric Polarization in Ceramic Lead Magnesium Niobate
,”
Appl. Phys. Lett.
,
79
(26), pp.
4420
4422
.10.1063/1.1426690
11.
Ma
,
W.
, and
Eric Cross
,
L.
,
2002
, “
Flexoelectric Polarization of Barium Strontium Titanate in the Paraelectric State
,”
Appl. Phys. Lett.
,
81
(18), pp.
3440
3442
.10.1063/1.1518559
12.
Ma
,
W.
, and
Eric Cross
,
L.
,
2003
, “
Strain-Gradient-Induced Electric Polarization in Lead Zirconate Titanate Ceramics
,”
Appl. Phys. Lett.
,
82
(19), pp.
3293
3295
.10.1063/1.1570517
13.
Ma
,
W.
, and
Eric Cross
,
L.
,
2005
, “
Flexoelectric Effect in Ceramic Lead Zirconate Titanate Ceramics
,”
Appl. Phys. Lett.
,
86
(7), p.
072905
.10.1063/1.1868078
14.
Ma
,
W.
, and
Eric Cross
,
L.
,
2006
, “
Flexoelectricity of Barium Titanate
,”
Appl. Phys. Lett.
,
88
(23), p.
232902
.10.1063/1.2211309
15.
Fu
,
J. Y.
,
Zhu
,
W.
,
Li
,
N.
, and
Eric Cross
,
L.
,
2006
, “
Experimental Studies of the Converse Flexoelectric Effect Induced by Inhomogeneous Electric Field in a Barium Strontium Titanate Composition
,”
J. Appl. Phys.
,
100
(2), p.
024112
.10.1063/1.2219990
16.
Cross
,
L.
,
2006
, “
Flexoelectric Effects: Charge Separation in Insulating Solids Subjected to Elastic Strain Gradients
,”
J. Mater. Sci.
,
41
(1), pp.
53
63
.10.1007/s10853-005-5916-6
17.
Maranganti
,
R.
,
Sharma
,
N. D.
, and
Sharma
,
P.
,
2006
, “
Electromechanical Coupling in Nonpiezoelectric Materials Due to Nanoscale Nonlocal Size Effects: Green's Function Solutions and Embedded Inclusions
,”
Phys. Rev. B
,
74
(1), p.
014110
.10.1103/PhysRevB.74.014110
18.
Sharma
,
N. D.
,
Maranganti
,
R.
, and
Sharma
,
P.
,
2007
, “
On the Possibility of Piezoelectric Nanocomposites Without Using Piezoelectric Materials
,”
J. Mech. Phys. Solids
,
55
(11), pp.
2328
2350
.10.1016/j.jmps.2007.03.016
19.
Ma
,
W.
,
2008
, “
A Study of Flexoelectric Coupling Associated Internal Electric Field and Stress in Thin Film Ferroelectrics
,”
Phys. Status Solidi B
,
245
(4), pp.
761
768
.10.1002/pssb.200743514
20.
Majdoub
,
M. S.
,
Sharma
,
P.
, and
Cagin
,
T.
,
2008
, “
Enhanced Size Dependent Piezoelectricity and Elasticity in Nanostructures Due to the Flexoelectric Effect
,”
Phys. Rev. B
,
77
(12), p.
125424
.10.1103/PhysRevB.77.125424
21.
Maranganti
,
R.
, and
Sharma
,
P.
,
2009
, “
Atomistic Determination of Flexoelectric Properties of Crystalline Dielectrics
,”
Phys. Rev. B
,
80
(5), p.
054109
.10.1103/PhysRevB.80.054109
22.
Ma
,
W.
,
2010
, “
Flexoelectric Charge Separation and Size Dependent Piezoelectricity in Dielectric Solids
,”
Phys. Status Solidi B
,
247
(
1
), pp.
213
218
.10.1002/pssb.200945394
23.
Hong
,
J.
,
Catalan
,
G.
,
Scott
,
J. F.
, and
Artacho
,
E.
,
2010
, “
The Flexoelectricity of Barium and Strontium Titanates From First Principles
,”
J. Phys.: Condens. Matter
,
22
(11), p.
112201
.10.1088/0953-8984/22/11/112201
24.
Gharbi
,
M.
,
Sun
,
Z. H.
,
Sharma
,
P.
,
White
,
K.
, and
El-Borgi
,
S.
,
2011
, “
Flexoelectric Properties of Ferroelectrics and the Nanoindentation Size-Effect
,”
Int. J. Solids Struct.
,
48
(
2
), pp.
249
256
.10.1016/j.ijsolstr.2010.09.021
25.
Baskaran
,
S.
,
Ramachandran
,
N.
,
He
,
X.
,
Thiruvannamalai
,
S.
, and
Lee
,
H. J.
,
2011
, “
Giant Flexoelectricity in Polyvinylidene Fluoride Films
,”
Phys. Lett. A
,
375
(20), pp.
2082
2084
.10.1016/j.physleta.2011.04.011
26.
Chu
,
B.
, and
Salem
,
D. R.
,
2012
, “
Flexoelectricity in Several Thermoplastic and Thermosetting Polymers
,”
Appl. Phys. Lett.
,
101
(10), p.
103905
.10.1063/1.4750064
27.
Yan
,
Z.
, and
Jiang
,
L.
,
2013
, “
Size-Dependent Bending and Vibration Behavior of Piezoelectric Nanobeams Due to Flexoelectricity
,”
J. Phys. D: Appl. Phys.
,
46
(35), p.
355502
.10.1088/0022-3727/46/35/355502
28.
Jiang
,
X.
,
Huang
,
W.
, and
Zhang
,
S.
,
2013
, “
Flexoelectric Nano-Generator: Materials, Structures and Devices
,”
Nano Energy
,
2
(6), pp.
1079
1092
.10.1016/j.nanoen.2013.09.001
29.
Hu
,
S.
,
Li
,
H.
, and
Tzou
,
H.
,
2013
, “
Flexoelectric Responses of Circular Rings
,”
ASME J. Vib. Acoust.
,
135
(2), p.
021003
.10.1115/1.4023044
30.
Yudin
,
P. V.
, and
Tagantsev
,
A. K.
, “
Fundamentals of Flexoelectricity in Solids
,”
Nanotechnology
,
24
(43), p.
432001
.10.1088/0957-4484/24/43/432001
31.
Zubko
,
P.
,
Catalan
,
G.
, and
Tagantsev
,
A. K.
,
2013
, “
Flexoelectric Effect in Solids
,”
Annu. Rev. Mater. Res.
,
43
, pp.
387
421
.10.1146/annurev-matsci-071312-121634
32.
Deng
,
Q.
,
Liu
,
L.
, and
Sharma
,
P.
,
2014
, “
Flexoelectricity in Soft Materials and Biological Membranes
,”
J. Mech. Phys. Solids
,
62
, pp.
209
227
.10.1016/j.jmps.2013.09.021
33.
Mohammadi
,
P.
,
Liu
,
L.
, and
Sharma
,
P.
,
2014
, “
A Theory of Flexoelectric Membranes and Effective Properties of Heterogeneous Membranes
,”
ASME J. Appl. Mech.
,
81
(
1
), p.
011007
.10.1115/1.4023978
34.
Shmakov
,
S. L.
,
2011
, “
A Universal Method of Solving Quartic Equations
,”
Int. J. Pure Appl. Math.
,
71
(2), pp.
251
259
.
You do not currently have access to this content.