Abstract

Recent work has highlighted how the phenomenon of flexoelectricity can masquerade as piezoelectricity. This notion can not only be exploited to create artificial piezoelectric-like materials without using piezoelectric materials but may also explain measurement artifacts in dielectrics. In this article, we show that the reverse is also possible and potentially advantageous in certain situations (such as energy harvesting). By constructing a computational homogenization approach predicated on the finite element method, we argue that composites made of piezoelectric phases can conspire to endow the material with a distinct overall flexoelectric-like response even though the native flexoelectricity of the constituent materials is negligible. Full finite element procedures for numerical evaluation of the different effective tensors, including the flexoelectric tensor, are provided. Numerical investigations are conducted, showing variation of the effective flexoelectric properties with respect to local geometry and properties of the composite in piezoelectric–piezoelectric and polymer–piezoelectric composites. We find that the flexoelectric response can be tuned to nearly five times higher than the constituents.

Issue Section:
Research Papers
Keywords:
computational homogenization, strain gradient, second-order homogenization, finite element, flexoelectricity, computational mechanics, mechanical properties of materials, micromechanics
Topics:
Composite materials, Piezoelectricity, Strain gradient, Tensors, Finite element methods, Finite element analysis

References

1.
Buchberger
,
G.
,
Schwödiauer
,
R.
, and
Bauer
,
S.
,
2008
, “
Flexible Large Area Ferroelectret Sensors for Location Sensitive Touchpads
,”
Appl. Phys. Lett.
,
92
(
12
), p.
123511
. 10.1063/1.2903711
2.
Carpi
,
F.
,
De Rossi
,
D.
,
Kornbluh
,
R.
,
Pelrine
,
R. E.
, and
Sommer-Larsen
,
P.
,
2011
,
Dielectric Elastomers as Electromechanical Transducers: Fundamentals, Materials, Devices, Models and Applications of an Emerging Electroactive Polymer Technology
,
Elsevier
,
Amsterdam
.
3.
Yang
,
D.
,
Verma
,
M. S.
,
So
,
J.-H.
,
Mosadegh
,
B.
,
Keplinger
,
C.
,
Lee
,
B.
,
Khashai
,
F.
,
Lossner
,
E.
,
Suo
,
Z.
, and
Whitesides
,
G. M.
,
2016
, “
Buckling Pneumatic Linear Actuators Inspired by Muscle
,”
Adv. Mater. Technol.
,
1
(
3
), p.
1600055
. 10.1002/admt.201600055
4.
Dagdeviren
,
C.
,
Yang
,
B. D.
,
Su
,
Y.
,
Tran
,
P. L.
,
Joe
,
P.
,
Anderson
,
E.
,
Xia
,
J.
,
Doraiswamy
,
V.
,
Dehdashti
,
B.
, and
Feng
,
X.
,
Lu
,
B.
,
Poston
,
R.
,
Khalpey
,
Z.
,
Ghaffari
,
R.
,
Huang
,
Y.
,
Slepian
,
M. J.
, and
Rogers
,
J. A
.,
2014
, “
Conformal Piezoelectric Energy Harvesting and Storage From Motions of the Heart, Lung, and Diaphragm
,”
Proc. Natl. Acad. Sci. USA
,
111
(
5
), pp.
1927
–
1932
. 10.1073/pnas.1317233111
5.
Bauer
,
S.
,
Bauer-Gogonea
,
S.
,
Graz
,
I.
,
Kaltenbrunner
,
M.
,
Keplinger
,
C.
, and
Schwödiauer
,
R.
,
2014
, “
25th Anniversary Article: A Soft Future: From Robots and Sensor Skin to Energy Harvesters
,”
Adv. Mater.
,
26
(
1
), pp.
149
–
162
. 10.1002/adma.201303349
6.
Huang
,
J.
,
Shian
,
S.
,
Suo
,
Z.
, and
Clarke
,
D. R.
,
2013
, “
Maximizing the Energy Density of Dielectric Elastomer Generators Using Equi-Biaxial Loading
,”
Adv. Funct. Mater.
,
23
(
40
), pp.
5056
–
5061
. 10.1002/adfm.201300402
7.
Yang
,
S.
,
Zhao
,
X.
, and
Sharma
,
P.
,
2017
, “
Avoiding the Pull-In Instability of a Dielectric Elastomer Film and the Potential for Increased Actuation and Energy Harvesting
,”
Soft. Matter.
,
13
, pp.
4552
–
4558
. 10.1039/C7SM00542C
8.
Carpi
,
F.
,
Bauer
,
S.
, and
De Rossi
,
D.
,
2010
, “
Stretching Dielectric Elastomer Performance
,”
Science
,
330
(
6012
), pp.
1759
–
1761
. 10.1126/science.1194773
9.
Rogers
,
J. A.
,
Someya
,
T.
, and
Huang
,
Y.
,
2010
, “
Materials and Mechanics for Stretchable Electronics
,”
Science
,
327
(
5973
), pp.
1603
–
1607
. 10.1126/science.1182383
10.
Muralt
,
P.
,
Polcawich
,
R. G.
, and
Trolier-McKinstry
,
S.
,
2009
, “
Piezoelectric Thin Films for Sensors, Actuators, and Energy Harvesting
,”
MRS Bull.
,
34
(
9
), pp.
658
–
664
. 10.1557/mrs2009.177
11.
Erturk
,
A.
, and
Inman
,
D. J.
,
2011
,
Piezoelectric Energy Harvesting
,
John Wiley & Sons
,
Hoboken, NJ
.
12.
Kim
,
H. S.
,
Kim
,
J.-H.
, and
Kim
,
J.
,
2011
, “
A Review of Piezoelectric Energy Harvesting Based on Vibration
,”
Int. J. Precis. Eng. Manuf.
,
12
(
6
), pp.
1129
–
1141
. 10.1007/s12541-011-0151-3
13.
Ueda
,
J.
,
Secord
,
T. W.
, and
Asada
,
H. H.
,
2009
, “
Large Effective-Strain Piezoelectric Actuators Using Nested Cellular Architecture With Exponential Strain Amplification Mechanisms
,”
IEEE/ASME Trans. Mechatronics
,
15
(
5
), pp.
770
–
782
. 10.1109/TMECH.2009.2034973
14.
Ihn
,
J.-B.
, and
Chang
,
F.-K.
,
2004
, “
Detection and Monitoring of Hidden Fatigue Crack Growth Using a Built-In Piezoelectric Sensor/Actuator Network: I. Diagnostics
,”
Smart Mater. Struct.
,
13
(
3
), p.
609
. 10.1088/0964-1726/13/3/020
15.
Uchino
,
K.
,
2008
, “
Piezoelectric Actuators 2006
,”
J. Electroceram.
,
20
(
3–4
), pp.
301
–
311
. 10.1007/s10832-007-9196-1
16.
Seminara
,
L.
,
Capurro
,
M.
,
Cirillo
,
P.
,
Cannata
,
G.
, and
Valle
,
M.
,
2011
, “
Electromechanical Characterization of Piezoelectric Pvdf Polymer Films for Tactile Sensors in Robotics Applications
,”
Sens. Actuators., A.
,
169
(
1
), pp.
49
–
58
. 10.1016/j.sna.2011.05.004
17.
Liu
,
L. P.
,
2014
, “
An Energy Formulation of Continuum Magneto-electro-elasticity With Applications
,”
J. Mech. Phys. Solids.
,
63
, pp.
451
–
480
. 10.1016/j.jmps.2013.08.001
18.
Choi
,
S.-B.
, and
Kim
,
G.-W.
,
2017
, “
Measurement of Flexoelectric Response in Polyvinylidene Fluoride Films for Piezoelectric Vibration Energy Harvesters
,”
J. Phys. D Appl. Phys.
,
50
(
7
), p.
075502
. 10.1088/1361-6463/50/7/075502
19.
Ma
,
W.
, and
Cross
,
L. E.
,
2006
, “
Flexoelectricity of Barium Titanate
,”
Appl. Phys. Lett.
,
88
(
23
), p.
232902
. 10.1063/1.2211309
20.
Fu
,
J. Y.
,
Zhu
,
W.
,
Li
,
N.
, and
Cross
,
L. E.
,
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
21.
Eliseev
,
E. A.
,
Morozovska
,
A. N.
,
Glinchuk
,
M. D.
, and
Blinc
,
R.
,
2009
, “
Spontaneous Flexoelectric/flexomagnetic Effect in Nanoferroics
,”
Phys. Rev. B
,
79
(
16
), p.
165433
. 10.1103/PhysRevB.79.165433
22.
Petrov
,
A. G.
,
2006
, “
Electricity and Mechanics of Biomembrane Systems: Flexoelectricity in Living Membranes
,”
Anal. Chim. Acta.
,
568
(
1
), pp.
70
–
83
. 10.1016/j.aca.2006.01.108
23.
Todorov
,
A. T.
,
Petrov
,
A. G.
, and
Fendler
,
J. H.
,
1994
, “
First Observation of the Converse Flexoelectric Effect in Bilayer Lipid Membranes
,”
J. Phys. Chem.
,
98
(
12
), pp.
3076
–
3079
. 10.1021/j100063a004
24.
Ahmadpoor
,
F.
, and
Sharma
,
P.
,
2015
, “
Flexoelectricity in Two-Dimensional Crystalline and Biological Membranes
,”
Nanoscale
,
7
(
40
), pp.
16555
–
16570
. 10.1039/C5NR04722F
25.
Krichen
,
S.
, and
Sharma
,
P.
,
2016
, “
Flexoelectricity: A Perspective on an Unusual Electromechanical Coupling
,”
ASME J. Appl. Mech.
,
83
(
3
), p.
030801
. 10.1115/1.4032378
26.
Yudin
,
P. V.
, and
Tagantsev
,
A. K.
,
2013
, “
Fundamentals of Flexoelextricity in Solids
,”
Nanotechnology
,
24
, p.
432001
. 10.1088/0957-4484/24/43/432001
27.
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
28.
Lee
,
D.
, and
Noh
,
T. W.
,
2012
, “
Giant Flexoelectric Effect Through Interfacial Strain Relaxation
,”
Philos. Trans. R. Soc. London A Math., Phys. Eng. Sci.
,
370
(
1977
), pp.
4944
–
4957
. 10.1098/rsta.2012.0200
29.
Mbarki
,
R.
,
Baccam
,
N.
,
Dayal
,
K.
, and
Sharma
,
P.
,
2014
, “
Piezoelectricity Above the Curie Temperature? Combining Flexoelectricity and Functional Grading to Enable High-temperature Electromechanical Coupling
,”
Appl. Phys. Lett.
,
104
(
12
), p.
122904
. 10.1063/1.4869478
30.
Mao
,
S.
, and
Purohit
,
P. K.
,
2014
, “
Insights Into Flexoelectric Solids From Strain-Gradient Elasticity
,”
ASME J. Appl. Mech.
,
81
(
8
), p.
081004
. 10.1115/1.4027451
31.
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
32.
Nguyen
,
T. D.
,
Mao
,
S.
,
Yeh
,
Y.-W.
,
Purohit
,
P. K.
, and
McAlpine
,
M. C.
,
2013
, “
Nanoscale Flexoelectricity
,”
Adv. Mater.
,
25
(
7
), pp.
946
–
974
. 10.1002/adma.201203852
33.
Deng
,
Q.
,
Kammoun
,
M.
,
Erturk
,
A.
, and
Sharma
,
P.
,
2014
, “
Nanoscale Flexoelectric Energy Harvesting
,”
Int. J. Solids. Struct.
,
51
(
18
), pp.
3218
–
3225
. 10.1016/j.ijsolstr.2014.05.018
34.
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
35.
Rahmati
,
A. H.
,
Bauer
,
S.
, and
Sharma
,
P.
,
2019
, “
Nonlinear Bending Deformation of Soft Electrets and Prospects for Engineering Flexoelectricity and Transverse (d 31) Piezoelectricity
,”
Soft. Matter.
,
15
(
1
), pp.
127
–
148
. 10.1039/C8SM01664J
36.
Wen
,
X.
,
Li
,
D.
,
Tan
,
K.
,
Deng
,
Q.
, and
Shen
,
S.
,
2019
, “
Flexoelectret: An Electret With a Tunable Flexoelectriclike Response
,”
Phys. Rev. Lett.
,
122
(
14
), p.
148001
. 10.1103/PhysRevLett.122.148001
37.
Chandratre
,
S.
, and
Sharma
,
P.
,
2012
, “
Coaxing Graphene to Be Piezoelectric
,”
Appl. Phys. Lett.
,
100
(
2
), p.
023114
. 10.1063/1.3676084
38.
Wang
,
B.
, and
Sharma
,
P.
,
2019
, “
Flexoelectricity as a Universal Mechanism for Energy Harvesting From Crumpling of Thin Sheets
,”
Phys. Rev. B
,
100
(
3
), p.
035438
. 10.1103/PhysRevB.100.035438
39.
Abdollahi
,
A.
,
Vásquez-Sancho
,
F.
, and
Catalan
,
G.
,
2018
, “
Piezoelectric Mimicry of Flexoelectricity
,”
Phys. Rev. Lett.
,
121
(
20
), p.
205502
. 10.1103/PhysRevLett.121.205502
40.
Cholleti
,
E. R.
,
2018
, “
A Review on 3D Printing of Piezoelectric Materials
,” IOP Conference Series: Materials Science and Engineering, Vol.
455
,
IOP Publishing
, p.
012046
.
41.
Guinovart-Sanjuán
,
D.
,
Merodio
,
J.
,
López-Realpozo
,
J. C.
,
Vajravelu
,
K.
,
Rodríguez-Ramos
,
R.
,
Guinovart-Díaz
,
R.
,
Bravo-Castillero
,
J.
, and
Sabina
,
F. J.
,
2019
, “
Asymptotic Homogenization Applied to Flexoelectric Rods
,”
Materials
,
12
(
2
), p.
232
. 10.3390/ma12020232
42.
Sidhardh
,
S.
, and
Ray
,
M. C.
,
2018
, “
Effective Properties of Flexoelectric Fiber-Reinforced Nanocomposite
,”
Mater. Today Commun.
,
17
, pp.
114
–
123
. 10.1016/j.mtcomm.2018.08.008
43.
Eremeyev
,
V. A.
,
Ganghoffer
,
J.-F.
,
Konopińska-Zmysłowska
,
V.
, and
Uglov
,
N. K.
,
2020
, “
Flexoelectricity and Apparent Piezoelectricity of a Pantographic Micro-bar
,”
Int. J. Eng. Sci.
,
149
, p.
103213
. 10.1016/j.ijengsci.2020.103213
44.
Mohammadi
,
P.
,
Liu
,
L. P.
, 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
45.
Chen
,
H. T.
,
Zhang
,
S. D.
,
Soh
,
A. K.
, and
Yin
,
W. Y.
,
2015
, “
Phase Field Modeling of Flexoelectricity in Solid Dielectrics
,”
J. Appl. Phys.
,
118
(
3
), p.
034106
. 10.1063/1.4926795
46.
Nanthakumar
,
S. S.
,
Zhuang
,
X.
,
Park
,
H. S.
, and
Rabczuk
,
T.
,
2017
, “
Topology Optimization of Flexoelectric Structures
,”
J. Mech. Phys. Solids.
,
105
, pp.
217
–
234
. 10.1016/j.jmps.2017.05.010
47.
Ghasemi
,
H.
,
Park
,
H. S.
, and
Rabczuk
,
T.
,
2018
, “
A Multi-Material Level Set-Based Topology Optimization of Flexoelectric Composites
,”
Comput. Methods Appl. Mech. Eng.
,
332
, pp.
47
–
62
. 10.1016/j.cma.2017.12.005
48.
Milton
,
G. W.
,
2017
,
Extending the Theory of Composites to Other Areas of Science
,
Bookbaby
,
Pennsauken Township, NJ
.
49.
Mao
,
S.
,
Purohit
,
P. K.
, and
Aravas
,
N.
,
2016
, “
Mixed Finite-Element Formulations in Piezoelectricity and Flexoelectricity
,”
Proc. R. Soc. A Math., Phys. Eng. Sci.
,
472
(
2190
), p.
20150879
. 10.1098/rspa.2015.0879
50.
Hamdia
,
K. M.
,
Ghasemi
,
H.
,
Zhuang
,
X.
,
Alajlan
,
N.
, and
Rabczuk
,
T.
,
2018
, “
Sensitivity and Uncertainty Analysis for Flexoelectric Nanostructures
,”
Comput. Methods Appl. Mech. Eng.
,
337
, pp.
95
–
109
. 10.1016/j.cma.2018.03.016
51.
Abdollahi
,
A.
,
Christian
,
P. C
,
Daniel
,
M. D.
,
Arroyo
,
M.
, and
Irene
,
A. I.
,
2014
, “
Computational Evaluation of the Flexoelectric Effect in Dielectric Solids
,”
J. Appl. Phys.
,
116
(
9
), p.
093502
. 10.1063/1.4893974
52.
Marshall
,
J.
, and
Dayal
,
K.
,
2014
, “
Atomistic-to-Continuum Multiscale Modeling With Long-Range Electrostatic Interactions in Ionic Solids
,”
J. Mech. Phys. Solids.
,
62
, pp.
137
–
162
. 10.1016/j.jmps.2013.09.025
53.
Yang
,
L.
, and
Dayal
,
K.
,
2011
, “
A Completely Iterative Method for the Infinite Domain Electrostatic Problem With Nonlinear Dielectric Media
,”
J. Comput. Phys.
,
230
(
21
), pp.
7821
–
7829
. 10.1016/j.jcp.2011.07.001
54.
Feng
,
M.-L.
, and
Wu
,
C.-C.
,
2001
, “
A Study of Three-Dimensional Four-Step Braided Piezo-Ceramic Composites by the Homogenization Method
,”
Compos. Sci. Technol.
,
61
(
13
), pp.
1889
–
1898
. 10.1016/S0266-3538(01)00090-2
55.
Brenner
,
R.
,
2009
, “
Numerical Computation of the Response of Piezoelectric Composites Using Fourier Transform
,”
Phys. Rev. B
,
79
(
18
), p.
184106
. 10.1103/PhysRevB.79.184106
56.
Mindlin
,
R. D.
,
1964
, “
Micro-structure in Linear Elasticity
,”
Arch. Rational Mech. Anal.
,
16
(
1
), pp.
51
–
78
. 10.1007/BF00248490
57.
Mindlin
,
R. D.
, and
Eshel
,
N. N.
,
1968
, “
On First Strain-Gradient Theories in Linear Elasticity
,”
Int. J. Solids. Struct.
,
4
(
1
), pp.
109
–
124
. 10.1016/0020-7683(68)90036-X
58.
Yvonnet
,
J.
,
Auffray
,
N.
, and
Monchiet
,
V.
,
2020
, “
Computational Second-Order Homogenization of Materials With Effective Anisotropic Strain Gradient Behavior
,”
Int. J. Solids. Struct.
,
191–192
, pp.
434
–
448
. 10.1016/j.ijsolstr.2020.01.006
59.
Gologanu
,
M.
,
Leblond
,
J.-B.
,
Perrin
,
G.
, and
Devaux
,
J.
,
1997
, “Recent Extensions of Gurson’s Model for Porous Ductile Metals,”
Continuum Micromechanics
,
Springer
,
New York
, pp.
61
–
130
.
60.
Forest
,
S.
,
1998
, “
Mechanics of Generalized Continua: Construction by Homogenizaton
,”
Le J. de Phys. IV
,
8
(
PR4
), pp.
Pr4
–
Pr39
.
61.
Forest
,
S.
,
Pradel
,
F.
, and
Sab
,
K.
,
2001
, “
Asymptotic Analysis of Heterogeneous Cosserat Media
,”
Int. J. Solids. Struct.
,
38
(
26–27
), pp.
4585
–
4608
. 10.1016/S0020-7683(00)00295-X
62.
Monchiet
,
V.
,
Auffray
,
N.
, and
Yvonnet
,
J.
,
2020
, “
Strain-Gradient Homogenization: A Bridge Between Asymptotic Expansion and Quadratic Boundary Condition Methods
,”
Mech. Mater.
,
143
, p.
103309
. 10.1016/j.mechmat.2019.103309
63.
Yvonnet
,
J.
,
2019
,
Computational Homogenization of Heterogeneous Materials With Finite Elements
,
Springer Nature
,
Cham, Switzerland
.
64.
Hu
,
S.
, and
Shen
,
S.
,
2009
, “
Electric Field Gradient Theory With Surface Effect for Nano-dielectrics
,”
Comput., Mater. Continua (CMC)
,
13
(
1
), p.
63
.
65.
Kouznetsova
,
V. G.
,
Geers
,
M. G. D.
, and
Brekelmans
,
W. A. M.
,
2002
, “
Multi-scale Constitutive Modeling of Heterogeneous Materials With Gradient Enhanced Computational Homogenization Scheme
,”
Int. J. Numer. Methods Eng.
,
54
, pp.
1235
–
1260
. 10.1002/nme.541
66.
Kaczmarczyk
,
L.
,
Pearce
,
C. J.
, and
Bićanić
,
N.
,
2008
, “
Scale Transition and Enforcement of Rve Boundary Conditions in Second-order Computational Homogenization
,”
Int. J. Numer. Methods Eng.
,
74
(
3
), pp.
506
–
522
. 10.1002/nme.2188
67.
Yvonnet
,
J.
, and
Liu
,
L. P.
,
2017
, “
A Numerical Framework for Modeling Flexoelectricity and Maxwell Stress in Soft Dielectrics at Finite Strains
,”
Comput. Methods Appl. Mech. Eng.
,
313
, pp.
450
–
482
. 10.1016/j.cma.2016.09.007
68.
Pettermann
,
E. Z.
, and
Suresh
,
S.
,
2000
, “
A Comprehensive Unit Cell Model: A Study of Coupled Effects in Piezoelectric 1–3 Composites
,”
Int. J. Solids. Struct.
,
37
(
39
), pp.
5447
–
5464
. 10.1016/S0020-7683(99)00224-3
Copyright © 2020 by ASME
You do not currently have access to this content.