Piezoelectrics and ferroelectrics have been widely used in modern industries because of their peculiar electromechanical coupling properties, quick response, and compact size. In this work, we give a comprehensive review of our works and others' works in the past decade on the multiscale computational mechanics methods for electromechanical coupling behavior of piezoelectrics and ferroelectrics. The methods are classified into three types based on their applicable scale (i.e., macroscopic methods, mesoscopic methods, and atomic-level methods). In macroscopic methods, we first introduce the basic linear finite element method and employ it to analyze the crack problems in piezoelectrics. Then, the nonlinear finite element methods are presented for electromechanically coupled deformation and the domain switching processes were simulated. Based on our developed nonlinear electromechanically coupled finite element method, the domain switching instability problem was specially discussed and a constrained domain-switching model was proposed to overcome it. To specially address the crack problem in piezoelectrics, we further proposed a meshless electromechanical coupling method for piezoelectrics. In mesoscopic methods, the phase field methods (PFM) were firstly presented and the simulation results on the defects effect and size effect of deformation in ferroelectrics were given. Then, to solve the computational complexity problem of PFM in polycrystals, we proposed an optimization-based computational method taking the interactions between grains in an Eshelby inclusion manner. The domain texture evolution process can be calculated, and the Taylor's rule of plasticity has been reproduced well by this optimization-based model. Alternatively, the domain switching in polycrystalline ferroelectrics can be simulated by a proposed Monte Carlo method, which treated domain switching as a stochastic process. In atomic-level methods, we firstly introduce the first-principles method to calculate polarization and studied the topological polarization and strain gradient effect in ferroelectrics. Then, we present a modified electromechanically coupled molecular dynamic (MD) method for ferroelectrics based on the shell model and investigated the size effect of electromechanical deformation in ferroelectric thin films and nanowires. Finally, we introduced our recently proposed novel atomic finite element method (AFEM), which has higher computational efficiency than the MD. The deformation as well as domain evolution processes in ferroelectrics calculated by AFEM were also presented. The development of electromechanically coupled computational mechanics methods at multiscale is greatly beneficial, not only to the deformation and fracture of piezoelectrics/ferroelectrics, but also to structural design and reliability analysis of smart devices in engineering.

References

References
1.
Xu
,
Y.
,
1991
,
Ferroelectric Materials and Their Applications
,
North-Holland
,
Amsterdam
.
2.
Jaffe
,
B.
,
Cook
,
W. R.
, and
Jaffe
,
H.
,
1971
,
Piezoelectric Ceramics
,
Academic
,
London
.
3.
Lines
,
M. E.
, and
Glass
,
A. M.
,
1977
,
Principles and Applications of Ferroelectrics and Related Materials
,
Clarendon
,
Oxford
.
4.
Otsuka
,
K.
, and
Ren
,
B.
,
2005
, “
Physical Metallurgy of Ti-Ni-Based Shape Memory Alloys
,”
Prog. Mater. Sci.
,
50
, pp.
511
678
.10.1016/j.pmatsci.2004.10.001
5.
Chen
,
P. J.
, and
Peerey
,
P. S.
,
1979
, “
One-Dimensional Dynamic Electromechanical Constitutive Relations of Ferroelectric Materials
,”
Acta Mech.
,
31
, pp.
231
241
.10.1007/BF01176851
6.
Chen
,
P. J.
,
1980
, “
Three Dimensional Dynamic Electromechanical Constitutive Relations for Ferroelectric Materials
,”
Int. J. Solids Struct.
,
16
, pp.
1059
1067
.10.1016/0020-7683(80)90063-3
7.
Bassiouny
,
E.
,
Ghaleb
,
A. F.
, and
Maugin
,
G. A.
,
1988
, “
Thermodynamical Formulation for Coupled Electromechanical Hysteresis Effects-I. Basic Equations
,”
Int. J. Eng. Sci.
,
26
(
12
), pp.
1279
1295
.10.1016/0020-7225(88)90047-X
8.
Cocks
,
C. F.
, and
McMeeking
,
R. M.
,
1999
, “
A Phenomenological Constitutive Law for the Behaviour of Ferroelectric Ceramics
,”
Ferroelectrics
,
228
, pp.
219
228
.10.1080/00150199908226136
9.
Kamlah
,
M.
, and
Tsakmakis
,
C.
,
1999
, “
Phenomenological Modeling of the Nonlinear Electromechanical Coupling in Ferroelectrics
,”
Int. J. Solids Struct.
,
36
, pp.
669
695
.10.1016/S0020-7683(98)00040-7
10.
Huber
,
J. E.
, and
Fleck
,
N. A.
,
2001
, “
Multi-axial Electrical Switching of a Ferroelectric: Theory Versus Experiment
,”
J. Mech. Phys. Solids
,
49
, pp.
785
811
.10.1016/S0022-5096(00)00052-1
11.
Landis
,
C. M.
,
2002
, “
Fully Coupled, Multi-axial, Symmetric Constitutive Laws for Polycrystalline Ferroelectric Ceramics
,”
J. Mech. Phys. Solids
,
50
, pp.
127
152
.10.1016/S0022-5096(01)00021-7
12.
Hwang
,
S. C.
,
Lynch
,
C. S.
, and
McMeeking
,
R. M.
,
1995
, “
Ferroelectric/Ferroelastic Interactions and a Polarization Switching Model
,”
Acta Metall. Mater.
,
43
(
5
), pp.
2073
2084
.10.1016/0956-7151(94)00379-V
13.
Hwang
,
S. C.
,
Huber
,
J. E.
,
McMeeking
,
R. M.
, and
Fleck
,
N. A.
,
1998
, “
The Simulation of Switching in Polycrystalline Ferroelectric Ceramics
,”
J. Appl. Phys.
,
83
(
3
), pp.
1530
1540
.10.1063/1.368219
14.
Chen
,
X.
,
Fang
,
D. N.
, and
Hwang
,
K. C.
,
1997
, “
Micromechanics Simulation of Ferroelectric Polarization Switching
,”
Acta Mater.
,
45
(
8
), pp.
3181
3189
.10.1016/S1359-6454(97)00008-6
15.
Lu
,
W.
,
Fang
,
D. N.
,
Li
,
C. Q.
, and
Hwang
,
K. C.
,
1999
, “
Nonlinear Electric-Mechanical Behavior and Micromechanics Modeling of Ferroelectric Domain Evolution
,”
Acta Mater.
,
47
, pp.
2913
2926
.10.1016/S1359-6454(99)00153-6
16.
Li
,
F. X.
, and
Rajapakse
,
R. K. N. D.
,
2007
, “
A Constrained Domain Switching Model for Polycrystalline Ferroelectric Ceramics: Part I—Model Formulation and Application to Tetragonal Materials
,”
Acta Mater.
,
55
, pp.
6472
6480
.10.1016/j.actamat.2007.08.002
17.
Li
,
F. X.
, and
Rajapakse
,
R. K. N. D.
,
2007
, “
A Constrained Domain Switching Model for Polycrystalline Ferroelectric Ceramics: Part II—Combined Switching and Application to Rhombohedral Materials
,”
Acta Mater.
,
55
, pp.
6481
6488
.10.1016/j.actamat.2007.08.003
18.
Park
,
S. B.
, and
Sun
,
C. T.
,
1995
, “
Fracture Criteria for Piezoelectric Ceramics
,”
J. Am. Ceram. Soc.
,
78
, pp.
1475
1480
.10.1111/j.1151-2916.1995.tb08840.x
19.
Zhang
,
T. Y.
, and
Tong
,
P.
,
1996
, “
Fracture Mechanics for a Mode III Crack in a Piezoelectric Material
,”
Int. J. Solids Struct.
,
33
, pp.
343
359
.10.1016/0020-7683(95)00046-D
20.
Zhang
,
T. Y.
, and
Qian
,
C. F.
,
1998
, “
Linear Electro-elastic Analysis of a Cavity or a Crack in a Piezoelectric Material
,”
Int. J. Solids Struct.
,
35
, pp.
2121
2149
.10.1016/S0020-7683(97)00168-6
21.
Zhang
,
T. Y.
,
Zhao
,
M. H.
, and
Tong
,
P.
,
2002
, “
Fracture of Piezoelectric Ceramics
,”
Adv. Appl. Mech.
,
38
, pp.
147
289
.10.1016/S0065-2156(02)80104-1
22.
Chen
,
Y. H.
, and
Lu
,
T. R.
,
2003
, “
Cracks and Fracture in Piezoelectrics
,”
Adv. Appl. Mech.
,
39
, pp.
121
215
.10.1016/S0065-2156(02)39003-3
23.
Zhang
,
T. Y.
, and
Gao
,
C. F.
,
2004
, “
Fracture Behavior of Piezoelectric Materials
,”
Theor. Appl. Fract. Mech.
,
41
, pp.
339
379
.10.1016/j.tafmec.2003.11.019
24.
Zhang
,
T. Y.
,
Zhao
,
M. H.
, and
Gao
,
C. F.
,
2005
, “
The Strip Dielectric Breakdown Model
,”
Int. J. Fract.
,
132
, pp.
311
327
.10.1007/s10704-005-2054-8
25.
Yang
,
W.
, and
Zhu
,
T.
,
1998
, “
Switch-Toughening of Ferroelectrics Subjected to Electric Fields
,”
J. Mech. Phys. Solids
,
46
(
2
), pp.
291
311
.10.1016/S0022-5096(97)00062-8
26.
Zhu
,
T.
, and
Yang
,
W.
,
1997
, “
Toughness Variation of Ferroelectrics by Polarization Switch Under Non-uniform Electric Field
,”
Acta Mater.
,
45
(
11
), pp.
4695
4702
.10.1016/S1359-6454(97)00123-7
27.
Mao
,
G. Z.
, and
Fang
,
D. N.
,
2004
, “
Fatigue Crack Growth Induced by Domain Switching Under Electromechanical Load in Ferroelectrics
,”
Theor. Appl. Fract. Mech.
,
41
, pp.
115
123
.10.1016/j.tafmec.2003.11.009
28.
Fang
,
D. N.
,
Zhang
,
Y. H.
, and
Mao
,
G. Z.
,
2011
, “
A COD Fracture Model of Ferroelectric Ceramics With Applications in Electric Field Induced Fatigue Crack Growth
,”
Int. J. Fract.
,
167
, pp.
211
220
.10.1007/s10704-010-9546-x
29.
Gao
,
H.
,
Zhang
,
T. Y.
, and
Tong
,
P.
,
1997
, “
Local and Global Energy Release Rate for an Electrically Yield Crack in a Piezoelectric Ceramic
,”
J. Mech. Phys. Solids
,
45
, pp.
491
510
.10.1016/S0022-5096(96)00108-1
30.
Gong
,
X.
, and
Suo
,
Z.
,
1996
, “
Reliability of Ceramic Multiplayer Actuators: A Nonlinear Finite Element Simulation
,”
J. Mech. Phys. Solids
,
44
(
5
), pp.
751
769
.10.1016/0022-5096(95)00026-7
31.
Hom
,
C. L.
, and
Shankar
,
N.
,
1996
, “
A Finite Element Method for Electrostrictive Ceramics
,”
Int. J. Solids Struct.
,
33
(
12
), pp.
1757
1779
.10.1016/0020-7683(95)00123-9
32.
Qi
,
H.
,
Fang
,
D. N.
, and
Yao
,
Z. H.
,
1997
, “
FEM Analysis of Electro-mechanical Coupling Effect of Piezoelectric Materials
,”
Comp. Mater. Sci.
,
8
, pp.
283
290
.10.1016/S0927-0256(97)00041-4
33.
Hwang
,
S. C.
, and
McMeeking
,
R. M.
,
1999
, “
A Finite Element Model of Ferroelastic Polycrystals
,”
Int. J. Solids Struct.
,
36
(
10
), pp.
1541
1556
.10.1016/S0020-7683(98)00051-1
34.
Li
,
F. X.
, and
Fang
,
D. N.
,
2004
, “
Simulations of Domain Switching in Ferroelectrics by a Three-Dimensional Finite Element Model
,”
Mech. Mater.
,
36
, pp.
959
973
.10.1016/j.mechmat.2003.01.001
35.
Chen
,
L. Q.
,
2002
, “
Phase-Field Models for Microstructure Evolution
,”
Annu. Rev. Mater. Res.
,
32
, pp.
113
140
.10.1146/annurev.matsci.32.112001.132041
36.
Wang
,
J.
,
Shi
,
S. Q.
,
Chen
,
L. Q.
,
Li
,
Y. L.
, and
Zhang
,
T. Y.
,
2004
, “
Phase Field Simulations of Ferroelectric/Ferroelastic Polarization Switching
,”
Acta Mater.
,
52
, pp.
749
764
.10.1016/j.actamat.2003.10.011
37.
Wang
,
J.
,
Li
,
Y. L.
,
Chen
,
L. Q.
, and
Zhang
,
T. Y.
,
2005
, “
The Effect of Mechanical Strains on the Ferroelectric and Dielectric Properties of a Model Single Crystal – Phase Field Simulation
,”
Acta Mater.
,
53
, pp.
2495
2507
.10.1016/j.actamat.2005.02.011
38.
Wang
,
J.
, and
Zhang
,
T. Y.
,
2006
, “
Size Effects in Epitaxial Ferroelectric Islands and Thin Films
,”
Phys. Rev. B
,
73
, p.
144107
.10.1103/PhysRevB.73.144107
39.
Wang
,
J.
, and
Zhang
,
T.-Y.
,
2007
, “
Phase Field Simulations of Polarization Switching-Induced Toughening in Ferroelectric Ceramics
,”
Acta Mater.
,
55
, pp.
2465
2477
.10.1016/j.actamat.2006.11.041
40.
Zhang
,
W.
, and
Bhattacharya
,
K.
,
2005
, “
A Computational Model of Ferroelectric Domains. Part I: Model Formulation and Domain Switching
,”
Acta Mater.
,
53
, pp.
185
198
.10.1016/j.actamat.2004.09.016
41.
Zhang
,
W.
, and
Bhattacharya
,
K.
,
2005
, “
A Computational Model of Ferroelectric Domains. Part II: Grain Boundaries and Defect Pinning
,”
Acta Mater.
,
53
, pp.
199
209
.10.1016/j.actamat.2004.09.015
42.
Song
,
Y. C.
,
Soh
,
A. K.
, and
Ni
,
Y.
,
2007
, “
Phase Field Simulation of Crack Tip Domain Switching in Ferroelectrics
,”
J. Phys. D: Appl. Phys.
,
40
, pp.
1175
1182
.10.1088/0022-3727/40/4/040
43.
Su
,
Y.
, and
Landis
,
C. M.
,
2007
, “
Continuum Thermodynamics of Ferroelectric Domain Evolution: Theory, Finite Element Implementation, and Application to Domain Wall Pinning
,”
J. Mech. Phys. Solids
,
55
, pp.
280
305
.10.1016/j.jmps.2006.07.006
44.
Dayal
,
K.
, and
Bhattacharya
,
K.
,
2007
, “
A Real-Space Non-local Phase-Field Model of Ferroelectric Domain Patterns in Complex Geometries
,”
Acta Mater.
,
55
, pp.
1907
1917
.10.1016/j.actamat.2006.10.049
45.
Zhang
,
Y. H.
,
Li
,
J. Y.
, and
Fang
,
D. N.
,
2010
, “
Oxygen-Vacancy-Induced Memory Effect and Large Recoverable Strain in a Barium Titanate Single Crystal
,”
Phys. Rev. B
,
82
, p.
064103
.10.1103/PhysRevB.82.064103
46.
Zhang
,
Y. H.
,
Li
,
J. Y.
, and
Fang
,
D. N.
,
2010
, “
Size Dependent Domain Configuration and Electric Field Driven Evolution in Ultrathin Ferroelectric Films: A Phase Field Investigation
,”
J. Appl. Phys.
,
107
, p.
034107
.10.1063/1.3298475
47.
Abdollahi
,
A.
, and
Arias
,
I.
,
2011
, “
Phase-Field Modeling of the Coupled Microstructure and Fracture Evolution in Ferroelectric Single Crystals
,”
Acta Mater.
,
59
, pp.
4733
4746
.10.1016/j.actamat.2011.03.030
48.
Abdollahi
,
A.
, and
Arias
,
I.
,
2012
, “
Numerical Simulation of Intergranular and Transgranular Crack Propagation in Ferroelectric Polycrystals
,”
Int. J. Fract.
,
174
, pp.
3
15
.10.1007/s10704-011-9664-0
49.
Abdollahi
,
A.
, and
Arias
,
I.
,
2012
, “
Phase-Field Modeling of Crack Propagation in Piezoelectric and Ferroelectric Materials With Different Electromechanical Crack Conditions
,”
J. Mech. Phys. Solids
,
60
, pp.
2100
2126
.10.1016/j.jmps.2012.06.014
50.
Li
,
W.
, and
Landis
,
C. M.
,
2011
, “
Nucleation and Growth of Domains Near Crack Tips in Single Crystal Ferroelectrics
,”
Eng. Fract. Mech.
,
78
, pp.
1505
1513
.10.1016/j.engfracmech.2011.01.002
51.
Tang
,
W.
,
Fang
,
D. N.
, and
Li
,
J. Y.
,
2009
, “
Two-Scale Micromechanics-Based Probabilistic Modeling of Domain Switching in Ferroelectric Ceramics
,”
J. Mech. Phys. Solids
,
57
, pp.
1683
1701
.10.1016/j.jmps.2009.07.004
52.
Li
,
F. X.
, and
Soh
,
A. K.
,
2010
, “
An Optimization-Based Computational Model for Domain Evolution in Polycrystalline Ferroelastics
,”
Acta Mater.
,
58
, pp.
2207
2215
.10.1016/j.actamat.2009.12.006
53.
Li
,
F. X.
,
Zhou
,
X. L.
, and
Soh
,
A. K.
,
2010
, “
An Optimization-Based “Phase Field” Model for Polycrystalline Ferroelectrics
,”
Appl. Phys. Lett.
,
96
, p.
152905
.10.1063/1.3377899
54.
Zhou
,
X. L.
, and
Li
,
F. X.
,
2011
, “
Simulations of Domain Evolution in Morphotropic Ferroelectric Ceramics Under Electromechanical Loading Using an Optimization-Based Model
,”
J. Appl. Phys.
,
109
, p.
084107
.10.1063/1.3569583
55.
Migoni
,
R.
,
Bilz
,
H.
, and
Bauerle
,
D.
,
1976
, “
Origin of Raman-Scattering and Ferroelectricity in Oxidic Perovskites
,”
Phys. Rev. Lett.
,
37
, pp.
1155
1158
.10.1103/PhysRevLett.37.1155
56.
Khatib
,
D.
,
Migoni
,
R.
,
Kugel
,
G. E.
, and
Godefroy
,
L.
,
1989
, “
Lattice-Dynamics of BaTiO3 in the Cubic Phase
,”
J. Phys.: Condens. Matter
,
1
, pp.
9811
9822
.10.1088/0953-8984/1/49/002
57.
Tinte
,
S.
,
Stachiotti
,
M. G.
,
Sepliarsky
,
M.
,
Migoni
,
R. L.
, and
Rodriguez
,
C. O.
,
1999
, “
Atomistic Modelling of BaTiO3 Based on First-Principles Calculations
,”
J. Phys.: Condens. Matter
,
11
, pp.
9679
9690
.10.1088/0953-8984/11/48/325
58.
Sepliarsky
,
M.
,
Phillpot
,
S. R.
,
Wolf
,
D.
,
Stachiotti
,
M. G.
, and
Migoni
,
R. L.
,
2001
, “
Long-Ranged Ferroelectric Interactions in Perovskite Superlattices
,”
Phys. Rev. B
,
64
, p.
060101
.10.1103/PhysRevB.64.060101
59.
Sepliarsky
,
M.
,
Asthagiri
,
A.
,
Phillpot
,
S. R.
,
Stachiotti
,
M. G.
, and
Migoni
,
R. L.
,
2005
, “
Atomic-Level Simulation of Ferroelectricity in Oxide Materials
,”
Curr. Opin. Solid State Mater. Sci.
,
9
, pp.
107
113
.10.1016/j.cossms.2006.05.002
60.
Shimada
,
T.
,
Wakahara
,
K.
,
Umeno
,
Y.
, and
Kitamura
,
T.
,
2008
, “
Shell Model Potential for PbTiO3 and Its Applicability to Surfaces and Domain Walls
,”
J. Phys.: Condens. Matter
,
20
, p.
325225
.10.1088/0953-8984/20/32/325225
61.
Zhang
,
Y. H.
,
Hong
,
J. W.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2009
, “
Molecular Dynamics Investigations on the Size-Dependent Ferroelectric Behavior of BaTiO3 Nanowires
,”
Nanotechnology
,
20
, p.
405703
.10.1088/0957-4484/20/40/405703
62.
Zhang
,
Y. H.
,
Hong
,
J. W.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2010
, “
Strain Effect on Ferroelectric Behaviors of BaTiO3 Nanowires: A Molecular Dynamics Study
,”
Nanotechnology
,
21
, p.
015701
.10.1088/0957-4484/21/1/015701
63.
Zhang
,
Y. H.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2011
, “
Stress-Induced Phase Transition and Deformation Behavior of BaTiO3 Nanowires
,”
J. Appl. Phys.
,
110
, p.
054109
.10.1063/1.3633267
64.
Zhang
,
Y. H.
,
Sang
,
Y. L.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2011
, “
Critical Thickness and the Size-Dependent Curie Temperature of BaTiO3 Nanofilms
,”
J. Comput. Theor. Nanosci.
,
8
, pp.
867
872
.10.1166/jctn.2011.1766
65.
Stachiotti
,
M. G.
, and
Sepliarsky
,
M.
,
2011
, “
Toroidal Ferroelectricity in PbTiO3 Nanoparticles
,”
Phys. Rev. Lett.
,
106
,
p
. 137601.10.1103/PhysRevLett.106.137601
66.
Cohen
,
R. E.
,
1992
, “
Origin of Ferroelectricity in Perovskite Oxides
,”
Nature
,
358
, pp.
136
138
.10.1038/358136a0
67.
Junquera
,
J.
, and
Ghosez
,
P.
,
2003
, “
Critical Thickness for Ferroelectricity in Perovskite Ultrathin Films
,”
Nature
,
422
, pp.
506
509
.10.1038/nature01501
68.
Naumov
,
I. I.
,
Bellaiche
,
L.
, and
Fu
,
H.
,
2004
, “
Unusual Phase Transitions in Ferroelectric Nanodisks and Nanorods
,”
Nature
,
432
, pp.
737
740
.10.1038/nature03107
69.
Geneste
,
G.
,
Bousquet
,
E.
,
Junquera
,
J.
, and
Ghosez
,
P.
,
2006
, “
Finite-Size Effects in BaTiO3 Nanowires
,”
Appl. Phys. Lett.
,
88
, p.
112906
.10.1063/1.2186104
70.
Pilania
,
G.
,
Alpay
,
S. P.
, and
Ramprasad
,
R.
,
2009
, “
Ab Initio Study of Ferroelectricity in BaTiO3 Nanowires
,”
Phys. Rev. B
,
80
, p.
014113
.10.1103/PhysRevB.80.014113
71.
Allik
,
H.
, and
Hughes
,
T. J. R.
,
1970
, “
Finite Element Method for Piezoelectric Vibration
,”
Int. J. Numer. Meth. Eng.
,
2
, pp.
151
157
.10.1002/nme.1620020202
72.
Hom
,
C. L.
, and
Shankar
,
N.
,
1995
, “
A Numerical Analysis of Relaxor Ferroelectric Multilayered Actuators and 2-2 Composite Arrays
,”
Smart Mater. Struct.
,
4
, pp.
266
273
.10.1088/0964-1726/4/4/011
73.
Kamlah
,
M.
, and
Bohle
,
U.
,
2001
, “
Finite Element Analysis of Piezoceramic Components Taking Into Account Ferroelectric Hysteresis Behavior
,”
Int. J. Solids Struct.
,
38
, pp.
605
633
.10.1016/S0020-7683(00)00055-X
74.
Chen
,
W.
, and
Lynch
,
C. S.
,
1999
, “
Finite Element Analysis of Cracks in Ferroelectric Ceramic Materials
,”
Eng. Fract. Mech.
,
64
, pp.
539
562
.10.1016/S0013-7944(99)00084-3
75.
Li
,
F. X.
, and
Rajapakse
,
R. K. N. D.
,
2008
, “
Nonlinear Finite Element Modeling of Polycrystalline Ferroelectrics Based on Constrained Domain Switching
,”
Comp. Mater. Sci.
,
44
(
2
), pp.
322
329
.10.1016/j.commatsci.2008.03.040
76.
Guo
,
X. H.
,
2004
, “
Study on Mechanical Behavior of Thin Films
”, Ph.D. thesis, Tsinghua University, Beijing, China.
77.
Guo
,
X. H.
, and
Fang
,
D. N.
,
2004
, “
Simulation of Interface Cracking in Piezoelectric Layers
,”
Int. J. Nonlin. Sci. Numer. Simul.
,
5
(
3
), pp.
235
242
.10.1515/IJNSNS.2004.5.3.235
78.
Guo
,
X. H.
,
Fang
,
D. N.
,
Soh
,
A. K.
,
Kim
,
H. C.
, and
Lee
,
J. J.
,
2006
, “
Analysis of Piezoelectric Ceramic Multilayer Actuators Based on an Electro-mechanical Coupled Meshless Method
,”
Acta Mech. Sinica
,
22
, pp.
34
39
.10.1007/s10409-005-0089-8
79.
Li
,
Y. L.
,
Hu
,
S. Y.
,
Liu
,
Z. K.
, and
Chen
,
L. Q.
,
2001
, “
Phase-Field Model of Domain Structures in Ferroelectric Thin Films
,”
Appl. Phys. Lett.
,
78
, pp.
3878
3880
.10.1063/1.1377855
80.
Li
,
Y. L.
,
Hu
,
S. Y.
,
Liu
,
Z. K.
, and
Chen
,
L. Q.
,
2002
, “
Effect of Electrical Boundary Conditions on Ferroelectric Domain Structures in Thin Films
,”
Appl. Phys. Lett.
,
81
, pp.
427
429
.10.1063/1.1492025
81.
Li
,
Y. L.
,
Hu
,
S. Y.
,
Liu
,
Z. K.
, and
Chen
,
L. Q.
,
2002
, “
Effect of Substrate Constraint on the Stability and Evolution of Ferroelectric Domain Structures in Thin Films
,”
Acta Mater.
,
50
, pp.
395
411
.10.1016/S1359-6454(01)00360-3
82.
Li
,
Y. L.
,
Cross
,
L. E.
, and
Chen
,
L. Q.
,
2005
, “
A Phenomenological Thermodynamic Potential for BaTiO3 Single Crystals
,”
J. Appl. Phys.
,
98
, p.
064101
.10.1063/1.2042528
83.
Zhao
,
X. F.
,
Soh
,
A. K.
, and
Li
,
L.
,
2010
, “
Influence of Dipole Defects on Polarization Switching in the Vicinity of a Crack in Relaxor Ferroelectrics
,”
Philos. Mag. Lett.
,
90
, pp.
251
260
.10.1080/09500831003630757
84.
Wang
,
Y. U.
,
Jin
,
Y. M.
, and
Khachaturyan
,
A. G.
,
2002
, “
Phase Field Microelasticity Theory and Modeling of Elastically and Structurally Inhomogeneous Solid
,”
J. Appl. Phys.
,
92
, pp.
1351
1360
.10.1063/1.1492859
85.
Choudhury
,
S.
,
Li
,
Y. L.
,
Krill
,
C. E.
, and
Chen
,
L. Q.
,
2005
, “
Phase-Field Simulation of Polarization Switching and Domain Evolution in Ferroelectric Polycrystals
,”
Acta Mater.
,
53
, pp.
5313
5321
.10.1016/j.actamat.2005.07.040
86.
Schrade
,
D.
,
Mueller
,
R.
,
Xu
,
B. X.
, and
Gross
,
D.
,
2007
, “
Domain Evolution in Ferroelectric Materials: A Continuum Phase Field Model and Finite Element Implementation
,”
Comput. Mech. Appl. Mech. Eng.
,
196
, pp.
4365
4374
.10.1016/j.cma.2007.05.010
87.
Wang
,
J.
, and
Zhang
,
T.-Y.
,
2008
, “
Phase Field Simulations of a Permeable Crack Parallel to the Original Polarization Direction in a Ferroelectric Mono-domain
,”
Eng. Fract. Mech.
,
75
, pp.
4886
4897
.10.1016/j.engfracmech.2008.06.025
88.
Zhang
,
Y. H.
,
Xu
,
R.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2012
, “
An Electromechanical Atomic-Scale Finite Element Method for Simulating Evolutions of Ferroelectric Nanodomains
,”
J. Mech. Phys. Solids
,
60
, pp.
1383
1399
.10.1016/j.jmps.2012.04.012
89.
Wang
,
X.
, and
Shen
,
Y.
,
1995
, “
Some Basic Theory for Thermal Magnetic Electric Elastic Media
,”
Chinese J. Appl. Mech.
,
I2
(
2
), pp.
28
39
(in Chinese).
90.
Sosa
,
H.
,
1991
, “
Plane Problem in Piezoelectric Media With Defects
,”
Int. J. Solids Struct.
,
28
(
4
), pp.
491
505
.10.1016/0020-7683(91)90061-J
91.
Ghandi
,
K.
, and
Hagood
,
N. W.
,
1996
, “
Nonlinear Finite Element Modeling of Phase Transitions in Electro-mechanically Coupled Material
,”
Proc. SPIE
,
2715
, pp.
121
140
.10.1117/12.240847
92.
Ghandi
,
K.
, and
Hagood
,
N. W.
,
1997
, “
A Hybrid Finite-Element Model for Phase Transition in Nonlinear Electro-mechanically Coupled Material
,”
Proc. SPIE
,
3039
, pp.
97
112
.10.1117/12.276529
93.
Chen
,
W.
, and
Lynch
,
C. S.
,
2001
, “
Multiaxial Constitutive Behavior of Ferroelastic Materials
,”
ASME J. Eng. Mater. Technol.
,
123
, pp.
169
175
.10.1115/1.1329874
94.
Fett
,
T.
, and
Munz
,
D.
,
2003
, “
Deformation of PZT Under Tension, Compression, Bending, and Torsion Loading
,”
Adv. Eng. Mater.
,
5
, pp.
718
722
.10.1002/adem.200300385
95.
Li
,
F. X.
, and
Fang
,
D. N.
,
2005
, “
Effects of Lateral Stress on the Electromechanical Response of Ferroelectric Ceramics: Experiments Versus Model
,”
J. Intell. Mater. Syst. Struct.
,
16
(
7–8
), pp.
583
588
.10.1177/1045389X05051627
96.
Steinkopff
,
T.
,
1999
, “
Finite-Element Modeling of Ferroic Domain Switching in Piezoelectric Ceramics
,”
J. Eur. Ceram. Soc.
,
19
, pp.
1247
1249
.10.1016/S0955-2219(98)00413-0
97.
Zeng
,
W.
,
Manzari
,
M. T.
,
Lee
,
J. D.
, and
Shen
,
Y. L.
,
2003
, “
Fully Coupled Non-linear Analysis of Piezoelectric Solids Involving Domain Switching
,”
Int. J. Numer. Meth. Eng.
,
56
, pp.
13
34
.10.1002/nme.556
98.
Kamlah
,
M.
,
Liskowsky
,
A. C.
,
McMeeking
,
R. M.
, and
Balke
,
H.
,
2005
, “
Finite Element Simulations of a Polycrystalline Ferroelectric Based on a Multidomain Single Crystal Switching Model
,”
Int. J. Solids Struct.
,
42
, pp.
2949
2964
.10.1016/j.ijsolstr.2004.09.045
99.
Fang
,
D. N.
, and
Li
,
C. Q.
,
1999
, “
Nonlinear Electric-Mechanical Behavior of a Soft PZT-51 Ferroelectric Ceramic
,”
J. Mater. Sci.
,
34
, pp.
4001
4010
.10.1023/A:1004603729657
100.
Axelsson
,
O.
,
1994
,
Iterative Solution Methods
,
Cambridge University
,
Cambridge, UK
.
101.
Crisfield
,
M. A.
,
1983
, “
An Arc-Length Method Including Line Searches and Accelerations
,”
Int. J. Numer. Meth. Eng.
,
19
, pp.
1269
1289
.10.1002/nme.1620190902
102.
Landis
,
C. M.
,
2002
, “
A New Finite-Element Formulation for Electromechanical Boundary Value Problems
,”
Int. J. Numer. Meth. Eng.
,
55
, pp.
613
628
.10.1002/nme.518
103.
Matthies
,
H.
, and
Strang
,
G.
,
1979
, “
The Solution of Non-linear Finite Element Equations
,”
Int. J. Numer. Meth. Eng.
,
14
, pp.
1613
1626
.10.1002/nme.1620141104
104.
Wempner
,
G. A.
,
1971
, “
Discrete Approximations Related to Nonlinear Theories of Solids
,”
Int. J. Solids Struct.
,
7
, pp.
1581
1599
.10.1016/0020-7683(71)90038-2
105.
Riks
,
E.
,
1972
, “
The Application of Newtons Method to the Problem of Elastic Instability
,”
J. Appl. Mech.
,
39
, pp.
1060
1066
.10.1115/1.3422829
106.
Kessler
,
H.
, and
Balke
,
H.
,
2001
, “
On the Local and Average Energy Release in Polarization Switching Phenomena
,”
J. Mech. Phys. Solids
,
49
, pp.
953
978
.10.1016/S0022-5096(00)00073-9
107.
Lucy
,
L. B.
,
1977
, “
A Numerical Approach to the Fission Hypothesis
,”
J. Astron.
,
8
(
12
), pp.
1013
1024
.10.1086/112164
108.
Monaghan
,
J. J.
,
1992
, “
Smoothed Particle Hydrodynamics
,”
Ann. Rev. Astron. Astrophys
,
30
, pp.
543
574
.10.1146/annurev.aa.30.090192.002551
109.
Nayroles
,
B.
,
Touzot
,
G.
, and
Villion
,
P.
,
1992
, “
Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Element
,”
Comput. Mech.
,
10
, pp.
307
318
.10.1007/BF00364252
110.
Belytschko
,
T.
,
Gu
,
L.
, and
Lu
,
Y. Y.
,
1994
, “
Fracture and Crack Growth by Element Free Galerkin Methods
,”
Model. Simul. Mater. Sci. Eng.
,
2
(
3A
), pp. 519–534.10.1088/0965-0393/2/3A/007
111.
Belytschko
,
T.
,
Lu
,
Y. Y.
, and
Gu
,
L.
,
1994
, “
Element-Free Galerkin Methods
,”
Int. J. Numer. Meth. Eng.
,
37
, pp.
229
256
.10.1002/nme.1620370205
112.
Belytschko
,
T.
,
Organ
,
D.
, and
Gerlach
,
C.
,
2000
, “
Element-Free Galerkin Methods for Dynamic Fracture in Concrete
,”
Comp. Meth. Appl. Mech. Eng.
,
187
, pp.
385
399
.10.1016/S0045-7825(00)80002-X
113.
Onate
,
E.
,
Idelsohn
,
S.
,
Zienkiewicaz
,
O. C.
, and
Taylor
,
R. L.
,
1996
, “
A Finite Point Method in Computational Mechanics: Applications to Convective Transport and Fluid Flow
,”
Int. J. Numer. Meth. Eng.
,
39
, pp.
3839
3866
.10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R
114.
Zhang
,
X.
,
Liu
,
X. H.
,
Song
,
K. Z.
, and
Lu
,
M. W.
,
2011
, “
Least-Square Collocation Meshless Method
,”
Int. J. Numer. Meth. Eng.
,
51
(
9
), pp.
1089
1100
.10.1002/nme.200
115.
Xu
,
T.
,
Zou
,
P.
,
Xu
,
T. S.
, and
Jiye
,
C. M.
,
2010
, “
Study on Weight Function of Meshless Method Based on B-Spline Wavelet Function
,”
The 3rd Int. Joint Conference on Computational Science and Optimization
, pp.
36
40
.
116.
Razmjoo
,
H.
,
Movahhedi
,
M.
, and
Hakimi
,
A.
,
2010
, “
Improved Meshless Method Using Direct Shape Function for Computational Electromagnetics
,”
Proceedings of the Asia-Pacific Microwave Conference
, Yokohama, Japan, pp.
2157
2160
.
117.
Sladek
,
J.
,
Sladek
,
V.
,
Solek
,
P.
, and
Pan
,
E.
,
2008
, “
Fracture Analysis of Cracks in Magneto-electro-elastic Solids by the MLPG
,”
Comput. Mech.
,
42
, pp.
697
714
.10.1007/s00466-008-0269-z
118.
Feng
,
W. J.
,
Han
,
X.
, and
Li
,
Y. S.
,
2009
, “
Fracture Analysis for Two-Dimensional Plane Problems of Nonhomogeneous Magneto-Electro-Thermo-Elastic Plates Subjected to Thermal Shock by Using the Messless Local Petrov Galerkin Method
,”
Computer Model. Eng. Sci.
,
48
(1), pp.
1
26
.
119.
Sosa
,
H. A.
, and
Pak
,
Y. E.
,
1990
, “
Three Dimensional Eigenfunction Analysis of a Crack in a Piezoelectric Material
,”
Int. J. Solids Struct.
,
26
, pp.
1
15
.10.1016/0020-7683(90)90090-I
120.
Li
,
Y. L.
, and
Chen
,
L. Q.
,
2006
, “
Temperature-Strain Phase Diagram for BaTiO3 Thin Films
,”
Appl. Phys. Lett.
,
88
, p.
072905
.10.1063/1.2172744
121.
Li
,
Y. L.
,
Choudhury
,
S.
,
Haeni
,
J. H.
,
Biegalski
,
M. D.
,
Vasudevarao
,
A.
,
Sharan
,
A.
,
Ma
,
H. Z.
,
Levy
,
J.
,
Gopalan
,
V.
,
Trolier-McKinstry
,
S.
,
Schlom
,
D. G.
,
Jia
,
Q. X.
, and
Chen
,
L. Q.
,
2006
, “
Phase Transitions and Domain Structures in Strained Pseudocubic (100) SrTiO3 Thin Films
,”
Phys. Rev. B
,
73
, p.
184112
.10.1103/PhysRevB.73.184112
122.
Liu
,
P.-L.
,
Wang
,
J.
,
Zhang
,
T.-Y.
,
Li
,
Y.
,
Chen
,
L.-Q.
,
Ma
,
X.-Q.
,
Chu
,
W.-Y.
, and
Qiao
,
L.-J.
,
2008
, “
Effects of Unequally Biaxial Misfit Strains on Polarization Phase Diagrams in Embedded Ferroelectric Thin Layers: Phase Field Simulations
,”
Appl. Phys. Lett.
,
93
, p.
132908
.10.1063/1.2975161
123.
Sheng
,
G.
,
Zhang
,
J. X.
,
Li
,
Y. L.
,
Choudhury
,
S.
,
Jia
,
Q. X.
,
Liu
,
Z. K.
, and
Chen
,
L. Q.
,
2008
, “
Misfit Strain-Misfit Strain Diagram of Epitaxial BaTiO(3) Thin Films: Thermodynamic Calculations and Phase-Field Simulations
,”
Appl. Phys. Lett.
,
93
, p.
232904
.10.1063/1.3039410
124.
Wang
,
J. J.
,
Wu
,
P. P.
,
Ma
,
X. Q.
, and
Chen
,
L. Q.
,
2010
, “
Temperature-Pressure Phase Diagram and Ferroelectric Properties of BaTiO(3) Single Crystal Based on a Modified Landau Potential
,”
J. Appl. Phys.
,
108
, p.
114105
.10.1063/1.3504194
125.
Wang
,
J.
, and
Kamlah
,
M.
,
2009
, “
Three-Dimensional Finite Element Modeling of Polarization Switching in a Ferroelectric Single Domain With an Impermeable Notch
,”
Smart Mater. Struct
.
18
, p.
104008
.10.1088/0964-1726/18/10/104008
126.
Xu
,
B.-X.
,
Schrade
,
D.
,
Gross
,
D.
, and
Mueller
,
R.
,
2010
, “
Phase Field Simulation of Domain Structures in Cracked Ferroelectrics
,”
Int. J. Fract.
,
165
, pp.
163
173
.10.1007/s10704-010-9471-z
127.
Miehe
,
C.
,
Welschinger
,
F.
, and
Hofacker
,
M.
,
2010
, “
A Phase Field Model of Electromechanical Fracture
,”
J. Mech. Phys. Solids
,
58
, pp.
1716
1740
.10.1016/j.jmps.2010.06.013
128.
Hong
,
L.
,
Soh
,
A. K.
,
Song
,
Y. C.
, and
Lim
,
L. C.
,
2008
, “
Interface and Surface Effects on Ferroelectric Nano-thin Films
,”
Acta Mater.
,
56
, pp.
2966
2974
.10.1016/j.actamat.2008.02.034
129.
Wang
,
J.
,
Kamlah
,
M.
,
Zhang
,
T. Y.
,
Li
,
Y.
, and
Chen
,
L. Q.
,
2008
, “
Size-Dependent Polarization Distribution in Ferroelectric Nanostructures: Phase Field Simulations
,”
Appl. Phys. Lett.
,
92
, p.
162905
.10.1063/1.2917715
130.
Wang
,
J.
,
Kamlah
,
M.
, and
Zhang
,
T.-Y.
,
2009
, “
Phase Field Simulations of Ferroelectric Nanoparticles With Different Long-Range-Electrostatic and -Elastic Interactions
,”
J. Appl. Phys.
,
105
, p.
014104
.10.1063/1.3043576
131.
Li
,
Y. L.
,
Hu
,
S. Y.
,
Tenne
,
D.
,
Soukiassian
,
A.
,
Schlom
,
D. G.
,
Chen
,
L. Q.
,
Xi
,
X. X.
,
Choi
,
K. J.
,
Eom
,
C. B.
,
Saxena
,
A.
,
Lookman
,
T.
, and
Jia
,
Q. X.
,
2007
, “
Interfacial Coherency and Ferroelectricity of BaTiO(3)/SrTiO(3) Superlattice Films
,”
Appl. Phys. Lett.
,
91
, p.
252904
.10.1063/1.2823608
132.
Xiao
,
Y.
,
Shenoy
, V
. B.
, and
Bhattacharya
,
K.
,
2005
, “
Depletion Layers and Domain Walls in Semiconducting Ferroelectric Thin Films
,”
Phys. Rev. Lett.
,
95
, p.
247603
.10.1103/PhysRevLett.95.247603
133.
Xiao
,
Y.
, and
Bhattacharya
,
K.
,
2008
, “
A Continuum Theory of Deformable, Semiconducting Ferroelectrics
,”
Arch. Ration. Mech. Anal.
,
189
, pp.
59
95
.10.1007/s00205-007-0096-y
134.
Hong
,
L.
,
Soh
,
A. K.
,
Du
,
Q. G.
, and
Li
,
J. Y.
,
2008
, “
Interaction of O Vacancies and Domain Structures in Single Crystal BaTiO3: Two-Dimensional Ferroelectric Model
,”
Phys. Rev. B
,
77
,
p
. 094104.10.1103/PhysRevB.77.094104
135.
Ren
,
X. B.
,
2004
, “
Large Electric-Field-Induced Strain in Ferroelectric Crystals by Point-Defect-Mediated Reversible Domain Switching
,”
Nature Mater.
,
3
, pp.
91
94
.10.1038/nmat1051
136.
Zhang
,
L. X.
, and
Ren
,
X.
,
2005
, “
In Situ Observation of Reversible Domain Switching in Aged Mn-Doped BaTiO3 Single Crystals
,”
Phys. Rev. B
,
71
, p.
174108
.10.1103/PhysRevB.71.174108
137.
Shu
,
Y. C.
, and
Bhattacharya
,
K.
,
2001
, “
Domain Patterns and Macroscopic Behaviour of Ferroelectric Materials
,”
Philos. Mag. B
,
81
, pp.
2021
2054
.10.1080/13642810108208556
138.
Devonshire
,
A. F.
,
1954
, “
Theory of Ferroelectrics
,”
Adv. Phys.
,
3
, pp.
85
130
.10.1080/00018735400101173
139.
Pertsev
,
N. A.
,
Zembilgotov
,
A. G.
, and
Tagantsev
,
A. K.
,
1998
, “
Effect of Mechanical Boundary Conditions on Phase Diagrams of Epitaxial Ferroelectric Thin Films
,”
Phys. Rev. Lett.
,
80
, pp.
1988
1991
.10.1103/PhysRevLett.80.1988
140.
Koukhar
, V
. G.
,
Pertsev
,
N. A.
, and
Waser
,
R.
,
2001
, “
Thermodynamic Theory of Epitaxial Ferroelectric Thin Films With Dense Domain Structures
,”
Phys. Rev. B
,
64
, p.
214103
.10.1103/PhysRevB.64.214103
141.
Shu
,
Y. C.
,
Yen
,
J. H.
,
Chen
,
H. Z.
,
Li
,
J. Y.
, and
Li
,
L. J.
,
2008
, “
Constrained Modeling of Domain Patterns in Rhombohedral Ferroelectrics
,”
Appl. Phys. Lett.
,
92
, p.
052909
.10.1063/1.2842385
142.
Li
,
J. Y.
, and
Liu
,
D.
,
2004
, “
On Ferroelectric Crystals With Engineered Domain Configurations
,”
J. Mech. Phys. Solids
,
52
, pp.
1719
1742
.10.1016/j.jmps.2004.02.011
143.
Ma
,
Y. F.
, and
Li
,
J. Y.
,
2007
, “
Magnetization Rotation and Rearrangement of Martensite Variants in Ferromagnetic Shape Memory Alloys
,”
Appl. Phys. Lett.
,
90
, p.
172504
.10.1063/1.2730752
144.
Li
,
L. J.
,
Yang
,
Y.
,
Shu
,
Y. C.
, and
Li
,
J. Y.
,
2010
, “
Continuum Theory and Phase-Field Simulation of Magnetoelectric Effects in Multiferroic Bismuth Ferrite
,”
J. Mech. Phys. Solids
,
58
, pp.
1613
1627
.10.1016/j.jmps.2010.07.006
145.
Hu
,
H. L.
, and
Chen
,
L. Q.
,
1997
, “
Computer Simulation of 90 deg Ferroelectric Domain Formation in Two-Dimensions
,”
Mater. Sci. Eng. A
,
238
, pp.
182
191
.10.1016/S0921-5093(97)00453-X
146.
Rabe
,
K. M.
,
Ahn
,
C. H.
, and
Triscone
,
J. M.
,
2007
,
Physics of Ferroelectrics: A Modern Perspective
,
Springer
,
Heidelberg
.
147.
Zhang
,
L. X.
, and
Ren
,
X. B.
,
2006
, “
Aging Behavior in Single-Domain Mn-Doped BaTiO3 Crystals: Implication for a Unified Microscopic Explanation of Ferroelectric Aging
,”
Phys. Rev. B
,
73
, p.
094121
.10.1103/PhysRevB.73.094121
148.
Zhang
,
L. X.
,
Erdem
,
E.
,
Ren
,
X. B.
, and
Eichel
,
R. A.
,
2008
, “
Reorientation of (Mn-Ti(’)-V-O(Center Dot Center Dot))(x) Defect Dipoles in Acceptor-Modified BaTiO3 Single Crystals: An Electron Paramagnetic Resonance Study
,”
Appl. Phys. Lett.
,
93
, p.
202901
.10.1063/1.3006327
149.
Erhart
,
P.
,
Eichel
,
R. A.
,
Traskelin
,
P.
, and
Albe
,
K.
,
2007
, “
Association of Oxygen Vacancies With Impurity Metal Ions in Lead Titanate
,”
Phys. Rev. B
,
76
, p.
174116
.10.1103/PhysRevB.76.174116
150.
Hooton
,
J. A.
, and
Merz
,
W. J.
,
1955
, “
Etch Patterns and Ferroelectric Domains in Batio3 Single Crystals
,”
Phys. Rev.
,
98
, pp.
409
413
.10.1103/PhysRev.98.409
151.
Zhong
,
W. L.
,
Wang
,
Y. G.
,
Zhang
,
P. L.
, and
Qu
,
B. D.
,
1994
, “
Phenomenological Study of the Size Effect on Phase-Transitions in Ferroelectric Particles
,”
Phys. Rev. B
,
50
, pp.
698
703
.10.1103/PhysRevB.50.698
152.
Sang
,
Y. L.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2008
, “
The Size and Strain Effects on the Electric-Field-Induced Domain Evolution and Hysteresis Loop in Ferroelectric BaTiO3 Nanofilms
,”
Comp. Mater. Sci.
,
44
, pp.
404
410
.10.1016/j.commatsci.2008.04.001
153.
Asada
,
T.
, and
Koyama
,
Y.
,
2007
, “
Ferroelectric Domain Structures Around the Morphotropic Phase Boundary of the Piezoelectric Material PbZr1-xTixO3
,”
Phys. Rev. B
,
75
, p.
214111
.10.1103/PhysRevB.75.214111
154.
Lupascu
,
D. C.
,
2004
,
Fatigue in Ferroelectric Ceramics and Related Issues
,
Springer-Verlag
,
Berlin
.
155.
Eshelby
,
J. D.
,
1957
, “
The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems
,”
Proc. R. Soc. A
,
241
, pp.
376
396
.10.1098/rspa.1957.0133
156.
Uchida
,
N.
, and
Ikeda
,
T.
,
1967
, “
Electrostriction in Perovskite-Type Ferroelectric Ceramics
,”
Jpn. J. Appl. Phys.
,
6
, pp.
1079
1088
.10.1143/JJAP.6.1079
157.
Huber
,
J. E.
,
Fleck
,
N. A.
, and
McMeeking
,
R. M.
,
1999
, “
A Crystal Plasticity Model for Ferroelectrics
,”
Ferroelectrics
,
228
, pp.
39
52
.10.1080/00150199908226124
158.
Li
,
F. X.
,
Fang
,
D. N.
, and
Soh
,
A. K.
,
2004
, “
An Analytical Axisymmetric Model for the Poling-History Dependent Behavior of Ferroelectric Ceramics
,”
Smart Mater. Struct.
,
13
, pp.
668
675
.10.1088/0964-1726/13/4/004
159.
Han
,
S. P.
,
1977
, “
A Global Convergent Method for Nonlinear Programming
,”
J. Optim. Theory Appl.
,
22
, pp.
297
309
.10.1007/BF00932858
160.
Bunge
,
H. J.
,
1982
,
Texture Analysis in Materials Science
,
Butterworth
,
Berlin
.
161.
Li
,
F. X.
, and
Rajapakse
,
R. K. N. D.
,
2007
, “
Analytically Saturated Domain Orientation Textures and Electromechanical Properties of Ferroelectric Ceramics Due to Electric/Mechanical Loading
,”
J. Appl. Phys.
,
101
, p.
054110
.10.1063/1.2645889
162.
Hoffmann
,
M. J.
,
Hammer
,
M.
,
Endriss
,
A.
, and
Lupascu
,
D. C.
,
2001
, “
Correlation Between Microstructure, Strain Behavior, and Acoustic Emission of Soft PZT Ceramics
,”
Acta Mater.
,
49
, pp.
1301
1310
.10.1016/S1359-6454(01)00025-8
163.
Li
,
F. X.
, and
Zhou
,
X. L.
,
2011
, “
Simulations of Gradual Domain-Switching in Polycrystalline Ferroelectrics Using an Optimization-Based, Multidomain-Grain Model
,”
Comput. Struct.
,
89
, pp.
1142
1147
.10.1016/j.compstruc.2010.11.002
164.
Li
,
J. Y.
,
Rogan
,
R. C.
,
Üstündag
,
E.
, and
Bhattacharya
,
K.
,
2005
, “
Domain Switching in Polycrystalline Ferroelectric Ceramics
,”
Nature Mater.
,
4
, pp.
776
781
.10.1038/nmat1485
165.
Webber
,
K. G.
,
Aulbach
,
E.
,
Key
,
T.
,
Marsilius
,
M.
,
Granzow
,
T.
, and
Rödel
,
J.
,
2009
, “
Temperature-Dependent Ferroelastic Switching of Soft Lead Zirconate Titanate
,”
Acta Mater.
,
57
, pp.
4614
4623
.10.1016/j.actamat.2009.06.037
166.
Li
,
Y. W.
,
Zhou
,
X. L.
, and
Li
,
F. X.
,
2010
, “
Temperature Dependent Mechanical Depolarization of Ferroelectric Ceramics
,”
J. Phys. D: Appl. Phys.
,
43
, p.
175501
.10.1088/0022-3727/43/17/175501
167.
Taylor
,
G. I.
,
1938
, “
Plastic Strain in Metals
,”
J. Inst. Met.
,
62
, pp.
307
324
.
168.
Noheda
,
B.
,
Cox
,
D. E.
,
Shirane
,
G.
,
Guo
,
R.
,
Jones
,
B.
, and
Cross
,
L. E.
,
2001
, “
Stability of the Monoclinic Phase in the Ferroelectric Perovskite PbZr1-xTixO3
,”
Phys. Rev. B
,
63
, p.
014103
.10.1103/PhysRevB.63.014103
169.
Zhou
,
D. Y.
,
Kamlah
,
M.
, and
Munz
,
D.
,
2005
, “
Effects of Uniaxial Prestress on the Ferroelectric Hysteretic Response of Soft PZT
,”
J. Eur. Ceram. Soc.
,
25
, pp.
425
432
.10.1016/j.jeurceramsoc.2004.01.016
170.
Li
,
Y. W.
, and
Li
,
F. X.
,
2010
, “
Large Anisotropy of Fracture Toughness in Mechanically Poled/Depoled Ferroelectric Ceramics
,”
Script. Mater.
,
62
, pp.
313
316
.10.1016/j.scriptamat.2009.11.032
171.
Berlincourt
,
D. A.
,
Cmolik
,
C.
, and
Jaffe
,
H.
,
1960
, “
Piezoelectric Properties of Polycrystalline Lead Titanate Zirconate Compositions
,”
Proc. Inst. Radio Eng.
,
48
, pp.
220
229
.10.1109/JRPROC.1960.287467
172.
Berlincourt
,
D. A.
, and
Krueger
,
H. H. A.
,
1959
, “
Domain Processes in Lead Titanate Zirconate and Barium Titanate Ceramics
,”
J. Appl. Phys.
,
30
, pp.
1804
1810
.10.1063/1.1735059
173.
King-Smith
,
R. D.
, and
Vanderbilt
,
D.
,
1993
, “
Theory of Polarization of Crystalline Solids
,”
Phys. Rev. B
,
47
, pp.
1651
1654
.10.1103/PhysRevB.47.1651
174.
Resta
,
R.
,
1993
, “
Macroscopic Electric Polarization as a Geometric Quantum Phase
,”
EPL
,
22
, pp.
133
138
.10.1209/0295-5075/22/2/010
175.
King-Smith
,
R. D.
, and
Vanderbilt
,
D.
,
1994
, “
First-Principles Investigation of Ferroelectricity in Perovskite Compounds
,”
Phys. Rev. B
,
49
, pp.
5828
5844
.10.1103/PhysRevB.49.5828
176.
Meyer
,
B.
, and
Vanderbilt
,
D.
,
2001
, “
Ab Initio Study of BaTiO3 and PbTiO3 Surfaces in External Electric Fields
,”
Phys. Rev. B
,
63
, p.
205426
.10.1103/PhysRevB.63.205426
177.
Hong
,
J. W.
,
Catalan
,
G.
,
Fang
,
D. N.
,
Artacho
,
E.
, and
Scott
,
J. F.
,
2010
, “
Topology of the Polarization Field in Ferroelectric Nanowires From First Principles
,”
Phys. Rev. B
,
81
, p.
172101
.10.1103/PhysRevB.81.172101
178.
Neaton
,
J. B.
, and
Rabe
,
K. M.
,
2003
, “
Theory of Polarization Enhancement in Epitaxial BaTiO3/SrTiO3 Superlattices
,”
Appl. Phys. Lett.
,
82
, pp.
1586
1588
.10.1063/1.1559651
179.
Bousquet
,
E.
,
Dawber
,
M.
,
Stucki
,
N.
,
Lichtensteiger
,
C.
,
Hermet
,
P.
,
Gariglio
,
S.
,
Triscone
,
J.-M.
, and
Ghosez
,
P.
,
2008
, “
Improper Ferroelectricity in Perovskite Oxide Artificial Superlattices
,”
Nature
,
452
, pp.
732
736
.10.1038/nature06817
180.
Hao
,
F.
,
Hong
,
J.
, and
Fang
,
D.
,
2011
, “
Size Effect of Elastic and Electromechanical Properties of BaTiO3 Films From First-Principles Method
,”
Integr. Ferroelectrics
,
124
, pp.
79
86
.10.1080/10584587.2011.573723
181.
Spanier
,
J. E.
,
Kolpak
,
A. M.
,
Urban
,
J. J.
,
Grinberg
,
I.
,
Lian
,
O. Y.
,
Yun
,
W. S.
,
Rappe
,
A. M.
, and
Park
,
H.
,
2006
, “
Ferroelectric Phase Transition in Individual Single-Crystalline BaTiO3 Nanowires
,”
Nano Lett.
,
6
, pp.
735
739
.10.1021/nl052538e
182.
Fu
,
H.
, and
Bellaiche
,
L.
,
2003
, “
Ferroelectricity in Barium Titanate Quantum Dots and Wires
,”
Phys. Rev. Lett.
,
91
, p.
257601
.10.1103/PhysRevLett.91.257601
183.
Benedek
,
N. A.
, and
Fennie
,
C. J.
,
2011
, “
Hybrid Improper Ferroelectricity: A Mechanism for Controllable Polarization-Magnetization Coupling
,”
Phys. Rev. Lett.
,
106
, p.
107204
.10.1103/PhysRevLett.106.107204
184.
Martin
,
R. M.
,
1974
, “
Comment on Calculations of Electric Polarization in Crystals
,”
Phys. Rev. B
,
9
, pp.
1998
1999
.10.1103/PhysRevB.9.1998
185.
Vanderbilt
,
D.
, and
King-Smith
,
R. D.
,
1993
, “
Electric Polarization as a Bulk Quantity and Its Relation to Surface Charge
,”
Phys. Rev. B
,
48
, pp.
4442
4455
.10.1103/PhysRevB.48.4442
186.
Resta
,
R.
,
1994
, “
Macroscopic Polarization in Crystalline Dielectrics: The Geometric Phase Approach
,”
Rev. Mod. Phys.
,
66
, pp.
899
915
.10.1103/RevModPhys.66.899
187.
Resta
,
R.
, and
Vanderbilt
,
D.
,
2007
, “
Theory of Polarization: A Modern Approach
,”
Physics of Ferroelectrics, Topics in Applied Physics
,
Springer
,
Berlin/Heidelberg
, pp.
31
68
.
188.
Spaldin
,
N. A.
, “
A Beginner's Guide to the Modern Theory of Polarization
,”
J. Solid State Chem.
(in press).
189.
Zhong
,
W.
,
King-Smith
,
R. D.
, and
Vanderbilt
,
D.
,
1994
, “
Giant LO-TO Splittings in Perovskite Ferroelectrics
,”
Phys. Rev. Lett.
,
72
, pp.
3618
3621
.10.1103/PhysRevLett.72.3618
190.
Hong
,
J.
, and
Fang
,
D.
,
2008
, “
Size-Dependent Ferroelectric Behaviors of BaTiO3 Nanowires
,”
Appl. Phys. Lett.
,
92
, p.
012906
.10.1063/1.2830662
191.
Hong
,
J.
, and
Fang
,
D.
,
2008
, “
Systematic Study of the Ferroelectric Properties of Pb(Zr0.5Ti0.5)O3 Nanowires
,”
J. Appl. Phys.
,
104
, p.
064118
.10.1063/1.2982090
192.
Wang
,
Z. Y.
,
Hu
,
J.
, and
Yu
,
M. F.
,
2006
, “
One-Dimensional Ferroelectric Monodomain Formation in Single Crystalline BaTiO3 Nanowire
,”
Appl. Phys. Lett.
,
89
, p.
263119
.10.1063/1.2425047
193.
Mermin
,
N. D.
,
1979
, “
The Topological Theory of Defects in Ordered Media
,”
Rev. Mod. Phys.
,
51
, pp.
591
648
.10.1103/RevModPhys.51.591
194.
Kogan
,
S. M.
,
1964
, “
Piezoelectric Effect During Inhomogeneous Deformation and Acoustic Scattering of Carriers in Crystals
,”
Sov. Phys. Solid State
,
5
, pp.
2069
2070
.
195.
Tagantsev
,
A. K.
,
1986
, “
Piezoelectricity and Flexoelectricity in Crystalline Dielectrics
,”
Phys. Rev. B
,
34
, pp.
5883
5889
.10.1103/PhysRevB.34.5883
196.
Cross
,
L. E.
,
2006
, “
Flexoelectric Effects: Charge Separation in Insulating Solids Subjected to Elastic Strain Gradients
,”
J. Mater. Sci.
,
41
, pp.
53
63
.10.1007/s10853-005-5916-6
197.
Catalan
,
G.
,
Noheda
,
B.
,
McAneney
,
J.
,
Sinnamon
,
L.
, and
Gregg
,
J.
,
2005
, “
Strain Gradients in Epitaxial Ferroelectrics
,”
Phys. Rev. B
,
72
, p.
020102
.10.1103/PhysRevB.72.020102
198.
Zubko
,
P.
,
Catalan
,
G.
,
Buckley
,
A.
,
Welche
,
P. R. L.
, and
Scott
,
J. F.
,
2007
, “
Strain-Gradient-Induced Polarization in SrTiO3 Single Crystals
,”
Phys. Rev. Lett.
,
99
, p.
167601
.10.1103/PhysRevLett.99.167601
199.
Ma
,
W.
,
2008
, “
A Study of Flexoelectric Coupling Associated Internal Electric Field and Stress in Thin Film Ferroelectrics
,”
Physica Status Solidi B
,
245
, pp.
761
768
.10.1002/pssb.200743514
200.
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
, p.
112201
.10.1088/0953-8984/22/11/112201
201.
Resta
,
R.
,
2010
, “
Towards a Bulk Theory of Flexoelectricity
,”
Phys. Rev. Lett.
,
105
, p.
127601
.10.1103/PhysRevLett.105.127601
202.
Hong
,
J.
, and
Vanderbilt
,
D.
,
2011
, “
First-Principles Theory of Frozen-Ion Flexoelectricity
,”
Phys. Rev. B
,
84
, p.
180101
.10.1103/PhysRevB.84.180101
203.
Lee
,
D.
,
Yoon
,
A.
,
Jang
,
S. Y.
,
Yoon
,
J.-G.
,
Chung
,
J.-S.
,
Kim
,
M.
,
Scott
,
J. F.
, and
Noh
,
T. W.
,
2011
, “
Giant Flexoelectric Effect in Ferroelectric Epitaxial Thin Films
,”
Phys. Rev. Lett.
,
107
, p.
057602
.10.1103/PhysRevLett.107.057602
204.
Catalan
,
G.
,
Lubk
,
A.
,
Vlooswijk
,
A. H. G.
,
Snoeck
,
E.
,
Magen
,
C.
,
Janssens
,
A.
,
Rispens
,
G.
,
Rijnders
,
G.
,
Blank
,
D. H. A.
, and
Noheda
,
B.
,
2011
, “
Flexoelectric Rotation of Polarization in Ferroelectric Thin Films
,”
Nature Mater.
,
10
, pp.
963
967
.10.1038/nmat3141
205.
Lu
,
H.
,
Bark
,
C.-W.
,
Esque De Los Ojos
,
D.
,
Alcala
,
J.
,
Eom
,
C. B.
,
Catalan
,
G.
, and
Gruverman
,
A.
,
2012
, “
Mechanical Writing of Ferroelectric Polarization
,”
Science
,
336
, pp.
59
61
.10.1126/science.1218693
206.
Zhou
,
H.
,
Hong
,
J.
,
Zhang
,
Y.
,
Li
,
F.
,
Pei
,
Y.
, and
Fang
,
D.
,
2012
, “
Flexoelectricity Induced Increase of Critical Thickness in Epitaxial Ferroelectric Thin Films
,”
Physica B: Condens. Matter
,
407
, pp.
3377
3381
.10.1016/j.physb.2012.04.041
207.
Zhou
,
H.
,
Hong
,
J.
,
Zhang
,
Y.
,
Li
,
F.
,
Pei
,
Y.
, and
Fang
,
D.
,
2012
, “
External Uniform Electric Field Removing the Flexoelectric Effect in Epitaxial Ferroelectric Thin Films
,”
EPL
,
99
, p.
47003
.10.1209/0295-5075/99/47003
208.
Maranganti
,
R.
, and
Sharma
,
P.
,
2009
, “
Atomistic Determination of Flexoelectric Properties of Crystalline Dielectrics
,”
Phys. Rev. B
,
80
, p.
054109
.10.1103/PhysRevB.80.054109
209.
Ma
,
W.
, and
Cross
,
L. E.
,
2006
, “
Flexoelectricity of Barium Titanate
,”
Appl. Phys. Lett.
,
88
, p.
232902
.10.1063/1.2211309
210.
Parker
,
C. B.
,
Maria
,
J.-P.
, and
Kingon
,
A. I.
,
2002
, “
Temperature and Thickness Dependent Permittivity of (Ba,Sr)TiO3 Thin Films
,”
Appl. Phys. Lett.
,
81
, pp.
340
342
.10.1063/1.1490148
211.
Sinnamon
,
L. J.
,
Bowman
,
R. M.
, and
Gregg
,
J. M.
,
2002
, “
Thickness-Induced Stabilization of Ferroelectricity in SrRuO3/Ba0.5Sr0.5TiO3/Au Thin Film Capacitors
,”
Appl. Phys. Lett.
,
81
, pp.
889
891
.10.1063/1.1496144
212.
Ramirez
,
F.
,
Heyliger
,
P. R.
, and
Pan
,
E.
,
2006
, “
Discrete Layer Solution to Free Vibrations of Functionally Graded Magneto-Electro-Elastic Plates
,”
Mech. Adv. Mater. Struct.
,
13
, pp.
249
266
.10.1080/15376490600582750
213.
Ahn
,
C. H.
,
Rabe
,
K. M.
, and
Triscone
,
J.-M.
,
2004
, “
Ferroelectricity at the Nanoscale: Local Polarization in Oxide Thin Films and Heterostructures
,”
Science
,
303
, pp.
488
491
.10.1126/science.1092508
214.
Fong
,
D. D.
,
Stephenson
,
G. B.
,
Streiffer
,
S. K.
,
Eastman
,
J. A.
,
Auciello
,
O.
,
Fuoss
,
P. H.
, and
Thompson
,
C.
,
2004
, “
Ferroelectricity in Ultrathin Perovskite Films
,”
Science
,
304
, pp.
1650
1653
.10.1126/science.1098252
215.
Lee
,
H. N.
,
Christen
,
H. M.
,
Chisholm
,
M. F.
,
Rouleau
,
C. M.
, and
Lowndes
,
D. H.
,
2005
, “
Strong Polarization Enhancement in Asymmetric Three-Component Ferroelectric Superlattices
,”
Nature
,
433
, pp.
395
399
.10.1038/nature03261
216.
Zhu
,
X. H.
,
Evans
,
P. R.
,
Byrne
,
D.
,
Schilling
,
A.
,
Douglas
,
C.
,
Pollard
,
R. J.
,
Bowman
,
R. M.
,
Gregg
,
J. M.
,
Morrison
,
F. D.
, and
Scott
,
J. F.
,
2006
, “
Perovskite Lead Zirconium Titanate Nanorings: Towards Nanoscale Ferroelectric ‘Solenoids'?
Appl. Phys. Lett.
,
89
, p.
129913
.10.1063/1.2347893
217.
Sepliarsky
,
M.
,
Stachiotti
,
M. G.
, and
Migoni
,
R. L.
,
1995
, “
Structural Instabilities in KTaO3 and KNbO3 Described by the Nonlinear Oxygen Polarizability Model
,”
Phys. Rev. B
,
52
, pp.
4044
4049
.10.1103/PhysRevB.52.4044
218.
Sepliarsky
,
M.
,
Stachiotti
,
M. G.
, and
Migoni
,
R. L.
,
1997
, “
Ferroelectric Soft Mode and Relaxation Behavior in a Molecular-Dynamics Simulation of KNbO3 and KTaO3
,”
Phys. Rev. B
,
56
, pp.
566
571
.10.1103/PhysRevB.56.566
219.
Sepliarsky
,
M.
,
Stachiotti
,
M. G.
, and
Migoni
,
R. L.
,
2005
, “
Surface Reconstruction and Ferroelectricity in PbTiO3 Thin Films
,”
Phys. Rev. B
,
72
, p.
014110
.10.1103/PhysRevB.72.014110
220.
Tinte
,
S.
, and
Stachiotti
,
M. G.
,
2001
, “
Surface Effects and Ferroelectric Phase Transitions in BaTiO3 Ultrathin Films
,”
Phys. Rev. B
,
64
, p.
235403
.10.1103/PhysRevB.64.235403
221.
Sepliarsky
,
M.
,
Phillpot
,
S. R.
,
Stachiotti
,
M. G.
, and
Migoni
,
R. L.
,
2002
, “
Ferroelectric Phase Transitions and Dynamical Behavior in KNbO3/KTaO3 Superlattices by Molecular-Dynamics Simulation
,”
J. Appl. Phys.
,
91
, pp.
3165
3171
.10.1063/1.1435826
222.
Sepliarsky
,
M.
, and
Tinte
,
S.
,
2009
, “
Dynamical Behavior of the Phase Transition of Strained BaTiO(3) From Atomistic Simulations
,”
Physica B
,
404
, pp.
2730
2732
.10.1016/j.physb.2009.06.079
223.
Sang
,
Y.-L.
,
Liu
,
B.
, and
Fang
,
D.-N.
,
2008
, “
Strain and Size Effects on Ferroelectric Properties of BaTiO3 Nanofilms
,”
Chin. Phys. Lett.
,
25
, pp.
1113
1116
.10.1088/0256-307X/25/3/083
224.
Tinte
,
S.
,
Stachiotti
,
M. G.
,
Phillpot
,
S. R.
,
Sepliarsky
,
M.
,
Wolf
,
D.
, and
Migoni
,
R. L.
,
2004
, “
Ferroelectric Properties of BaxSr1-xTiO3 Solid Solutions Obtained by Molecular Dynamics Simulation
,”
J. Phys.: Condens. Matter
,
16
, pp.
3495
3506
.10.1088/0953-8984/16/20/019
225.
Wolf
,
D.
,
Keblinski
,
P.
,
Phillpot
,
S. R.
, and
Eggebrecht
,
J.
,
1999
, “
Exact Method for the Simulation of Coulombic Systems by Spherically Truncated, Pairwise r(-1) Summation
,”
J. Chem. Phys.
,
110
, pp.
8254
8282
.10.1063/1.478738
226.
Angoshtari
,
A.
, and
Yavari
,
A.
,
2011
, “
Convergence Analysis of the Wolf Method for Coulombic Interactions
,”
Phys. Lett. A
,
375
, pp.
1281
1285
.10.1016/j.physleta.2011.01.048
227.
Ewald
,
P. P.
,
1921
, “
Die Berechnung Optischer und Elektrostatischer Gitterpotentiale
,”
Ann. Phys.
,
369
, pp.
253
287
.10.1002/andp.19213690304
228.
Deleeuw
,
S. W.
,
Perram
,
J. W.
, and
Smith
,
E. R.
,
1980
, “
Simulation of Electrostatic Systems in Periodic Boundary-Conditions 1. Lattice Sums and Dielectric-Constants
,”
Proc. R. Soc. A
,
373
, pp.
27
56
.10.1098/rspa.1980.0135
229.
Padilla
,
J.
, and
Vanderbilt
,
D.
,
1997
, “
Ab Initio Study of BaTiO3 Surfaces
,”
Phys. Rev. B
,
56
, pp.
1625
1631
.10.1103/PhysRevB.56.1625
230.
Strukov
,
B. A.
,
Davitadze
,
S. T.
,
Taraskin
,
S. A.
,
Goltzman
,
B. M.
,
Shulman
,
S. G.
, and
Lemanov
,
V. V.
,
2003
, “
Thermodynamical Properties of the Thin Polycrystalline BaTiO3 Films on Substrates
,”
Ferroelectrics
,
286
, pp.
967
972
.10.1080/00150190390206428
231.
Drezner
,
Y.
, and
Berger
,
S.
,
2005
, “
Thermodynamic Stability of BaTiO3 Nano-domains
,”
Mater. Lett.
,
59
, pp.
1598
1602
.10.1016/j.matlet.2004.08.044
232.
Zhang
,
Y. H.
,
Hong
,
J. W.
,
Liu
,
B.
, and
Fang
,
D. N.
,
2010
, “
A Surface-Layer Model of Ferroelectric Nanowire
,”
J. Appl. Phys.
,
108
, p.
124109
.10.1063/1.3525267
233.
Liu
,
B.
,
Huang
,
Y.
,
Jiang
,
H.
,
Qu
,
S.
, and
Hwang
,
K. C.
,
2004
, “
The Atomic-Scale Finite Element Method
,”
Comput. Mech. Appl. Mech. Eng.
,
193
, pp.
1849
1864
.10.1016/j.cma.2003.12.037
234.
Liu
,
B.
,
Jiang
,
H.
,
Huang
,
Y.
,
Qu
,
S.
,
Yu
,
M. F.
, and
Hwang
,
K. C.
,
2005
, “
Atomic-Scale Finite Element Method in Multiscale Computation With Applications to Carbon Nanotubes
,”
Phys. Rev. B
,
72
, p.
035435
.10.1103/PhysRevB.72.035435
235.
Zhang
,
X.
,
Hashimoto
,
T.
, and
Joy
,
D. C.
,
1992
, “
Electron Holographic Study of Ferroelectric Domain Walls
,”
Appl. Phys. Lett.
,
60
, pp.
784
786
.10.1063/1.106519
236.
Merz
,
W. J.
,
1954
, “
Domain Formation and Domain Wall Motions in Ferroelectric BaTiO3 Single Crystals
,”
Phys. Rev.
,
95
, pp.
690
698
.10.1103/PhysRev.95.690
237.
Floquet
,
N.
, and
Valot
,
C.
,
1999
, “
Ferroelectric Domain Walls in BaTiO3: Structural Wall Model Interpreting Fingerprints in XRPD Diagrams
,”
Ferroelectrics
,
234
, pp.
107
122
.10.1080/00150199908225285
238.
Floquet
,
N.
,
Valot
,
C. M.
,
Mesnier
,
M. T.
,
Niepce
,
J. C.
,
Normand
,
L.
,
Thorel
,
A.
, and
Kilaas
,
R.
,
1997
, “
Ferroelectric Domain Walls in BaTiO3: Fingerprints in XRPD Diagrams and Quantitative HRTEM Image Analysis
,”
J. Phys. III
,
7
, pp.
1105
1128
.
239.
Hlinka
,
J.
, and
Marton
,
P.
,
2006
, “
Phenomenological Model of a 90 deg Domain Wall in BaTiO3-Type Ferroelectrics
,”
Phys. Rev. B
,
74
, p.
104104
.10.1103/PhysRevB.74.104104
240.
Tsou
,
N. T.
,
Potnis
,
P. R.
, and
Huber
,
J. E.
,
2011
, “
Classification of Laminate Domain Patterns in Ferroelectrics
,”
Phys. Rev. B
,
83
, p.
184120
.10.1103/PhysRevB.83.184120
241.
Pilania
,
G.
, and
Ramprasad
,
R.
,
2010
, “
Complex Polarization Ordering in PbTiO3 Nanowires: A First-Principles Computational Study
,”
Phys. Rev. B
,
82
, p.
155442
.10.1103/PhysRevB.82.155442
242.
Jiang
,
B.
,
Bai
,
Y.
,
Chu
,
W. Y.
,
Su
,
Y. J.
, and
Qiao
,
L. J.
,
2008
, “
Direct Observation of Two 90 Degrees Steps of 180 Degrees Domain Switching in BaTiO3 Single Crystal Under an Antiparallel Electric Field
,”
Appl. Phys. Lett.
,
93
, p.
152905
.10.1063/1.3000634
243.
Dieguez
,
O.
, and
Vanderbilt
,
D.
,
2006
, “
First-Principles Calculations for Insulators at Constant Polarization
”,
Phys. Rev. Lett.
,
96
, p.
056401
.10.1103/PhysRevLett.96.056401
244.
Stengel
,
M.
,
Spaldin
,
N. A.
, and
Vanderbilt
,
D.
,
2009
, “
Electric Displacement as the Fundamental Variable in Electronic-Structure Calculations
,”
Nature Phys.
,
5
, pp.
304
308
.10.1038/nphys1185
245.
Hong
,
J.
, and
Vanderbilt
,
D.
,
2011
, “
Mapping the Energy Surface of PbTiO3 in Multidimensional Electric-Displacement Space
,”
Phys. Rev. B
,
84
, p.
115107
.10.1103/PhysRevB.84.115107
246.
Hong
,
J.
, and
Vanderbilt
,
D.
,
2013
, “
Electrically Driven Octahedral Rotations in SrTiO3 and PbTiO3
,”
Phys. Rev. B
,
87
, p.
064104
.10.1103/PhysRevB.87.064104
247.
McCash
,
K.
,
Srikanth
,
A.
, and
Ponomareva
,
I.
,
2012
, “
Competing Polarization Reversal Mechanisms in Ferroelectric Nanowires
,”
Phys. Rev. B
,
86
, p.
214108
.10.1103/PhysRevB.86.214108
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