The flexoelectric effect is an electromechanical phenomenon that is universally present in all dielectrics and exhibits a strong size-dependency. Through a judicious exploitation of scale effects and symmetry, flexoelectricity has been used to design novel types of structures and materials including piezoelectric materials without using piezoelectric. Flexoelectricity links electric polarization with strain gradients and is rather difficult to estimate experimentally. One well-acknowledged approach is to fabricate truncated pyramids and/or cones and examine their electrical response. A theoretical model is then used to relate the measured experimental response to estimate the flexoelectric properties. In this work, we revisit the typical model that is used in the literature and solve the problem of a truncated cone under compression or tension. We obtained closed-form analytical solutions to this problem and examine the size and shape effects of flexoelectric response of the aforementioned structure. In particular, we emphasize the regime in which the existing models are likely to incur significant error.
Issue Section:
Research Papers
References
1.
Wang
, X.
, Song
, J.
, Zhang
, F.
, He
, C.
, Hu
, Z.
, and Wang
, Z.
, 2010
, “Electricity Generation Based on One-Dimensional Group-III Nitride Nanomaterials
,” Adv. Mater.
, 22
(19
), pp. 2155
–2158
.2.
Madden
, J. D. W.
, Vandesteeg
, N. A.
, Anquetil
, P. A.
, Madden
, P. G. A.
, Takshi
, A.
, Pytel
, R. Z.
, Lafontaine
, S. R.
, Wieringa
, P. A.
, and Hunter
, I. W.
, 2004
, “Artificial Muscle Technology: Physical Principles and Naval Prospects
,” IEEE Oceanic Eng. Soc.
, 29
(3
), pp. 706
–728
.3.
Gautschi
, G.
, 2002
, Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and Acoustic Emission Sensors, Materials and Amplifiers
, Springer-Verlag
, Berlin
.4.
Labanca
, M.
, Azzola
, F.
, Vinci
, R.
, and Rodella
, L. F.
, 2008
, “Piezoelectric Surgery: Twenty Years of Use
,” Br. J. Oral Maxillofac. Surg.
, 46
(4
), pp. 265
–269
.5.
Tadigadapa
, S.
, and Mateti
, K.
, 2009
, “Piezoelectric MEMS Sensors: State-of-the-Art and Perspectives
,” Meas. Sci. Technol.
, 20
(9
), p. 092001
.6.
Tagantsev
, A. K.
, 1986
, “Piezoelectricity and Flexoelectricity in Crystalline Dielectrics
,” Phys. Rev. B
, 34
(8
), pp. 5883
–5889
.7.
Tagantsev
, A. K.
, Meunier
, V.
, and Sharma
, P.
, 2009
, “Novel Electromechanical Phenomena at the Nanoscale: Phenomenological Theory and Atomistic Modeling
,” MRS Bull.
, 34
(9
), pp. 643
–647
.8.
Ahmadpoor
, F.
, and Sharma
, P.
, 2015
, “Flexoelectricity in Two-Dimensional Crystalline and Biological Membranes
,” Nanoscale
, 7
(40
), pp. 16555
–16570
.9.
Krichen
, S.
, and Sharma
, P.
, 2016
, “Flexoelectricity: A Perspective on an Unusual Electromechanical Coupling
,” ASME J. Appl. Mech.
, 83
(3
), p. 030801
.10.
Yudin
, P. V.
, and Tagantsev
, A. K.
, 2013
, “Fundamentals of Flexoelectricity in Solids
,” Nanotechnology
, 24
(43
), p. 432001
.11.
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
, p. 014110
.12.
Ma
, W.
, and Cross
, L. E.
, 2001
, “Large Flexoelectric Polarization in Ceramic Lead Magnesium Niobate
,” Appl. Phys. Lett.
, 79
(19
), pp. 4420
–4422
.13.
Ma
, W.
, and Cross
, L. E.
, 2002
, “Flexoelectric Polarization in Barium Strontium Titanate in the Paraelectric State
,” Appl. Phys. Lett.
, 81
(19
), pp. 3440
–3442
.14.
Ma
, W.
, and Cross
, L. E.
, 2003
, “Strain-Gradient-Induced Electric Polarization in Lead Zirconate Titanate Ceramics
,” Appl. Phys. Lett.
, 82
(19
), pp. 3293
–3295
.15.
Meyer
, R. B.
, 1969
, “Piezoelectric Effects in Liquid Crystals
,” Phys. Rev. Lett.
, 22
(18
), p. 918
.16.
Baskaran
, S.
, He
, X.
, Wang
, Y.
, and Fu
, J. Y.
, 2012
, “Strain Gradient Induced Electric Polarization in α-Phase Polyvinylidene Fluoride Films Under Bending Conditions
,” J. Appl. Phys.
, 111
(1
), p. 014109
.17.
Chu
, B.
, and Salem
, D. R.
, 2012
, “Flexoelectricity in Several Thermoplastic and Thermosetting Polymers
,” Appl. Phys. Lett.
, 101
(10
), p. 103905
.18.
Petrov
, A. G.
, 1975
, “Flexoelectric Model for Active Transport,” Physical and Chemical Bases of Biological Information Transfer
, Plenum Press
, New York
.19.
Deng
, Q.
, Liu
, L.
, and Sharma
, P.
, 2014
, “Flexoelectricity in Soft Materials and Biological Membranes
,” J. Mech. Phys. Solids
, 62
, pp. 209
–227
.20.
Ahmadpoor
, F.
, Deng
, Q.
, Liu
, L.
, and Sharma
, P.
, 2013
, “Apparent Flexoelectricity in Lipid Bilayer Membranes Due to External Charge and Dipolar Distributions
,” Phys. Rev. E
, 88
(5
), p. 050701
.21.
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
.22.
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
.23.
Cross
, L. E.
, 2006
, “Flexoelectric Effects: Charge Separation in Insulating Solids Subjected to Elastic Strain Gradients
,” J. Mater. Sci.
, 41
(1
), pp. 53
–63
.24.
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
.25.
Fu
, J. Y.
, Zhu
, W.
, Li
, N.
, Smith
, N. B.
, and Cross
, L. E.
, 2007
, “Gradient Scaling Phenomenon in Microsize Flexoelectric Piezoelectric Composites
,” Appl. Phys. Lett.
, 91
(18
), p. 182910
.26.
Zhu
, W.
, Fu
, J. Y.
, Li
, N.
, and Cross
, L. E.
, 2006
, “Piezoelectric Composites Based on the Enhanced Flexoelectric Effects
,” Appl. Phys. Lett.
, 89
(19
), p. 192904
.27.
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
.28.
Shen
, S.
, and Hu
, S.
, 2010
, “A Theory of Flexoelectricity With Surface Effect for Elastic Dielectrics
,” J. Mech. Phys. Solids
, 58
(5
), pp. 665
–677
.29.
Liping
, L.
, 2014
, “An Energy Formulation of Continuum Magneto-Electro-Elasticity With Applications
,” J. Mech. Phys. Solids
, 63
, pp. 451
–480
.30.
Maranganti
, R.
, and Sharma
, P.
, 2009
, “Atomistic Determination of Flexoelectric Properties of Crystalline Dielectrics
,” Phys. Rev. B
, 80
(5
), p. 054109
.31.
Sharma
, N. D.
, Landis
, C. M.
, and Sharma
, P.
, 2010
, “Piezoelectric Thin-Film Superlattices Without Using Piezoelectric Materials
,” J. Appl. Phys.
, 108
(2
), p. 024304
.32.
Sharma
, N. D.
, Landis
, C. M.
, and Sharma
, P.
, 2012
, “Erratum: Piezoelectric Thin-Film Super-Lattices Without Using Piezoelectric Materials
,” J. Appl. Phys.
, 111
(5
), p. 059901
.33.
Kalinin
, S. V.
, and Meunier
, V.
, 2008
, “Electronic Flexoelectricity in Low-Dimensional Systems
,” Phys. Rev. B
, 77
(3
), p. 033403
.34.
Fousek
, J.
, Cross
, L. E.
, and Litvin
, D. B.
, 1999
, “Possible Piezoelectric Composites Based on the Flexoelectric Effect
,” Mater. Lett.
, 39
(5
), pp. 287
–291
.35.
Chandratre
, S.
, and Sharma
, P.
, 2012
, “Coaxing Graphene to Be Piezoelectric
,” Appl. Phys. Lett.
, 100
(2
), p. 023114
.36.
Lee
, D.
, Yang
, S. M.
, Yoon
, J. G.
, and Noh
, T. W.
, 2012
, “Flexoelectric Rectification of Charge Transport in Strain-Graded Dielectrics
,” Nano Lett.
, 12
(12
), pp. 6436
–6440
.37.
Deng
, Q.
, Kammoun
, M.
, Erturk
, A.
, and Sharma
, P.
, 2014
, “Nanoscale Flexoelectric Energy Harvesting
,” Int. J. Solids Struct.
, 51
(18
), pp. 3218
–3225
.38.
Bhaskar
, U. K.
, Banerjee
, N.
, Abdollahi
, A.
, Wang
, Z.
, Schlom
, D. G.
, Rijnders
, G.
, and Catalan
, G.
, 2016
, “A Flexoelectric Microelectromechanical System on Silicon
,” Nat. Nanotechnol.
, 11
(3
), pp. 263
–266
.39.
Gharbi
, M.
, Sun
, Z. H.
, Sharma
, P.
, and White
, K.
, 2009
, “The Origins of Electromechanical Indentation Size Effect in Ferroelectrics
,” Appl. Phys. Lett.
, 95
(14
), p. 142901
.40.
Gharbi
, M.
, Sun
, Z. H.
, Sharma
, P.
, White
, K.
, and El-Borgi
, S.
, 2011
, “Flexoelectric Properties of Ferroelectrics and Nanoindentation Size-Effect
,” Int. J. Solids Struct.
, 48
(2
), pp. 249
–256
.41.
Lu
, J.
, Lv
, J.
, Liang
, X.
, Xu
, M.
, and Shen
, S.
, 2016
, “Improved Approach to Measure the Direct Flexoelectric Coefficient of Bulk Polyvinylidene Fluoride
,” J. Appl. Phys.
, 119
(9
), p. 094104
.42.
Abdollahi
, A.
, Millán
, D.
, Peco
, C.
, Arroyo
, M.
, and Arias
, I.
, 2015
, “Revisiting Pyramid Compression to Quantify Flexoelectricity: A Three-Dimensional Simulation Study
,” Phys. Rev. B
, 91
(10
), p. 104103
.43.
Lam
, D. C. C.
, Yang
, F.
, Chong
, A. C. M.
, Wang
, J.
, and Tong
, P.
, 2003
, “Experiments and Theory in Strain Gradient Elasticity
,” J. Mech. Phys. Solids
, 51
(8
), pp. 1477
–1508
.44.
Mindlin
, R. D.
, 1968
, “Polarization Gradient in Elastic Dielectrics
,” Int. J. Solids. Struct.
, 4
(6
), pp. 637
–642
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