A Micro Electro Mechanical Model for Polarization Swithing of Ferroelectric Materials

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A MICRO-ELECTRO-MECHANICAL MODEL FOR

POLARIZATION SWITCHING OF FERROELECTRIC

MATERIALS

W. CHEN and C. S. LYNCH{

School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

(Received 16 September 1997; accepted 5 June 1998)

Abstract ÐFerroelectric and ferroelastic switching are the major source of nonlinearity and hysteresis in fer-roelectric materials subjected to high electric ®eld or mechanical stress. A computational micromechanics

model for polycrystalline ferroelectric ceramics is developed based on consideration of the constitutivebehavior of single crystals. This model simulates the tetragonal and the rhombohedral crystal structures.Saturation of the linear piezoelectric eect is included. Interaction between dierent grains in the polycrys-talline ceramic is considered. A switching criterion is developed that accounts for dierent energy levels as-sociated with 908 and 1808 switching for the tetragonal structure (or 70.58, 109.58, and 1808 for therhombohedral structure). Experimental results on 8/65/35 PLZT are simulated and a parametric study of the eects of crystal structure, intergranular interaction, and phase transformation is performed. # 1998Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Electro-active ceramics are widely used as sensors

and actuators. This class of materials includes

piezoelectrics, ferroelectrics, electrostrictors and

phase changers. Applications include active damp-ing, precision positioning, structural control, and

noise control. Many applications require the largest

possible strain and force. This results in their oper-

ation at high stress and electric ®elds, in which the

electro-active ceramics show strong nonlinearity

and hysteresis. These properties can be seen in the

electric displacement vs electric ®eld curves and

strain vs electric ®eld curves measured on 8/65/35

lead lanthannina zirconate titanate (PLZT)

ceramics [1]. The hysteresis loops are the result of

ferroelectric switching. When the electric ®eld is

higher than the coercive ®eld (E c), the polarization

direction is reoriented. A ferroelastic material is onein which the direction of polarization can be reor-

iented by an applied stress ®eld. For a 8/35/65

PLZT, when the stress level is higher than the coer-

cive stress (sc), polarization switching occurs.

Nonlinearity and hysteresis may limit the perform-

ance of electro-ceramic actuators.

Very large strain materials such as phase chan-

ging lead lanthanum stannate zirconate titanate

(PLSnZT) [2] ceramics and lead zinc niobate (PZN)

single crystals [3] are under development for large

strain actuator applications, but nonlinearity and

hysteresis as well as intergranular cracking in cer-

amics are often associated with the larger strain. In

antiferroelectric PLSnZT, application of strong elec-

tric ®elds induces a ferroelectric phase [2]. PZN

single crystals change their structures from rhombo-

hedral to tetragonal, or vice versa in response to

large electric ®eld [3]. The coupling of stress andelectric ®eld is suciently complicated so that it is

dicult to accurately describe the multiaxial nonli-

nearity and hysteresis at the macroscopic level. A

framework for the nonlinear theory of ferroelectrics

has been developed based on the general thermo-

dynamics theory [4±7]. Recently, a relatively simple

macroscopic constitutive model was developed [8],

but it is limited to one-dimensional uniaxial load-

ing. Micro-electro-mechanics models give insight

into the phenomena that give rise to hysteresis and,

with this understanding, may give insight into how

to control hysteresis and cracking phenomena.

In previous micro-electro-mechanics work,

Hwang et al . [9] used a Preisach hysteresis model tosimulate each grain of a ceramic. A work energy

criterion was used to determine the critical loading

level at which polarization switching occurs. The

contribution of each grain to the macroscopic strain

and polarization of the ceramic was calculated by

averaging over all grains. A similar approach was

used by Chan and Hagood [10]. The previous

models did not consider several phenomena.

Models based on a tetragonal crystal structure were

used to simulate rhombohedral materials. The

dierences between the tetragonal structure and

rhombohedral structure are considered in this work.

In this work, dierent energy levels associated with908 and 1808 switching for the tetragonal structure

(or 70.58, 109.58, and 1808 switching for the rhom-

Acta mater. Vol. 46, No. 15, pp. 5303±5311, 1998# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/98 $19.00 + 0.00PII: S1359-6454(98)00207-9

{To whom all correspondence should be addressed.

5303



bohedral structure) are considered. These energy

levels could be obtained by ®tting the experimental

results on ferroelectrics. The model presented here

has extended previous models by considering tetra-gonal and rhombohedral crystal structures, local

constitutive laws for individual grains, dierent

switching criteria for 908 and 1808 switching, inter-

action eects between grains, and phase transitions

driven by electric ®eld and mechanical stress.

The development begins with consideration of the

rhombohedral and tetragonal perovskite type crys-

tal structures of ferroelectric single crystals. The

constitutive behavior of single crystals governs the

behavior of ceramics. At small ®eld levels the con-

stitutive behavior is linear but not isotropic, which

is governed by piezoelectric constitutive laws. At

high ®eld levels in the polarization direction, the

piezoelectric eect saturates. At high ®eld levels in

other directions, the directions of spontaneous

polarization and strain can be switched by an

applied electric ®eld or mechanical stress [1,11].

Ferroelectric and ferroelastic polarization switching

together with linear eects are the source of the

butter̄ y shaped strain vs electric ®eld curves and

the corresponding electric displacement vs electric

®eld loops [12]. Phase changes between the tetra-

gonal and rhombohedral structures are considered

in this model. Other phase changes such as antifer-

roelectric-to-ferroelectric (AF±F) could readily be

implemented.

Ceramics are modeled as many randomlyoriented grains. A global coordinate system is used

to describe the applied loads and local coordinate

systems are used to describe the crystallographic

orientation of each grain. The local coordinate sys-

tems are ®xed relative to the global coordinate sys-

tem. The possible polarization directions are de®ned

relative to the local coordinate system. When the

polarization direction switches, local piezoelectric

and elastic tensors are rotated to a new direction.

Each grain has single crystal anisotropic non-linear

constitutive behavior. The response of each grain is

computed in local coordinates. The macroscopic re-

sponse of the ceramic is computed from the volume

average of the local response of each grain.

Polarization switching does not occur homoge-

neously in single crystals. The mechanism is the

nucleation and growth of domains, and the mobility

of domain walls has a large eect on the constitu-

tive behavior of the crystal. This model smears

smaller length scale eects such as domain wall

motion into the single crystal constitutive behavior.

Grains interact with each other and with the glob-

ally applied stress and electric ®elds. The intergra-

nular interaction is modeled using a simpli®ed

approximation of an inclusion in a matrix, with the

matrix having the volume average properties of the

other grains.The model can be used to explore the eects of

physical properties and interactions on the micro-

mechanical scale, with observed macroscopic ma-

terial properties. Micromechanical properties

include intergranular interactions, texture in the

granular orientations, crystal structure eects,switching between structures (as is believed to occur

in PZT compositions near morphotropic bound-

aries), and constitutive behavior such as ferroelec-

trics with a saturating piezoelectric eect, quadratic

electrostriction with saturation and structural phase

changes with associated volume changes.

This work begins by simulating the results of a

series of experiments performed on ceramic rhom-

bohedral 8/65/35 PLZT. This is followed by a para-

metric study that explores the eects of

intergranular interaction, crystal structure, and

phase transformation on the hysteresis loops of fer-

roelectric ceramics.

2. MODEL DESCRIPTION

2.1. Crystal structure

Development of this model begins with consider-

ation of the rhombohedral and tetragonal perovs-

kite type crystal structures for ferroelectric grains.

The tetragonal and the rhombohedral crystal struc-

tures allow the polarization to occur toward six

face centers and the eight corners of the unit cell.

Under small loads, the polarization of the grain

undergoes a reversible change which is proportional

to the loading level in one of two ways: (i) an

applied stress deforms the crystal structure, result-

ing in a relative displacement of the positive and

negative ions; (ii) an applied electric ®eld changes

the relative displacement of the positive and nega-

tive ions, inducing deformation of the crystal. These

are the linear piezoelectric eects. At high ®eld

levels in the polarization direction, the piezoelectric

eect saturates. At high ®eld levels in other direc-

tions, the central ion is moved to another possible

site, and the direction of spontaneous polarization

is switched.

2.2. Constitutive behavior of single crystals

The individual grains are modeled as linear piezo-electric with ferroelectric and ferroelastic switching,

and saturation of the piezoelectric eect at high

®eld levels. This constitutive behavior is described

by

eij eeij e

pij es

ij 1

Dm Ddm Dp

m DsmX 2

Superscripts e, p, and d indicate elastic, piezoelec-

tric, and dielectric components. esij is the spon-

taneous strain, and Dsm the spontaneous electric

displacement.

2.2.1. Linear constitutive behavior. The elastic,dielectric, and piezoelectric components, with stress

and electric ®eld as free variables, are described by

CHEN and LYNCH: SWITCHING OF FERROELECTRIC MATERIALS5304



the following equations:

elij ee

ij epij sijkl skl d nij E n 3

Dlm Dp

m Ddm d mkl skl emnE n 4

where superscript l refers to linear, sijkl is the elastic

compliance, d nij the piezoelectricity, E n the electric

®eld, and emn the dielectric permittivity. Indices

vary from 1 to 3. Summation on repeated indices is

implied.

Since the constitutive behavior is implemented at

the single crystal scale, and general tensor calcu-

lations are performed, anisotropic crystal properties

are readily implemented. Simulations presented in

this work are performed with the elastic compliance

and dielectric permittivity considered isotropic, and

the piezoelectricity tensor transversely isotropic. Inthis case, sijkl and emn are given by

sijkl 1 #

Y dikd jl À

#

Y dij dkl 5

emn edmn 6

where Y is Young's modulus, n is Poisson's ratio,

and e the dielectric permittivity.

The piezoelectricity tensor has a principal direc-

tion that aligns with the polarization direction in

each grain. The nonzero components of the piezo-

electric tensor include d 113, d 223, d 311, d 322, and d 333.

The x3-axis, parallel to the polarization direction, isthe transverse isotropic symmetric axis of the piezo-

electricity tensor. Polarization reorientation results

in a direction change of the anisotropic elastic,

dielectric, and piezoelectricity tensors. When the

polarization direction switches, the principal direc-

tions of the piezoelectricity tensor are rotated to

those associated with the new polarization direc-

tion.

2.2.2. Saturation of the linear piezoelectric eect.

Experimental data indicate that when a large

applied electric ®eld is in the direction of polariz-

ation, the linear piezoelectric eect saturates. This is

described by

epij

& d nij E an if e

pij esa

ij

esaij if e

pij b esa

ij

7

where esaij is the magnitude of the saturation strain,

E an are the components of the applied electric ®eld.

2.2.3. Ferroelectric and ferroelastic switching.

When the electric or stress ®eld is higher than the

coercive ®eld and it is not in the polarization direc-

tion, ferroelectric or ferroelastic switching occurs.

The magnitude of spontaneous strain and electric

displacement remain constant during switching, but

their directions change. One of the principle direc-

tions of strain is always aligned with the polariz-ation and the magnitude of strain in this direction

is positive, i.e. elongation. Non-1808 polarization

reorientation, through either ferroelectric or ferroe-

lastic switching, results in a change of the spon-

taneous strain. This is a major source of

nonlinearity and hysteresis.2.2.4. Switching between structures (TET±RH,

AF±F). Phase changes between the tetragonal and

rhombohedral structures are known to occur in

some materials. To accommodate this behavior, the

electric ®eld and/or mechanical stress are allowed to

drive the structures from tetragonal to rhombohe-

dral, or vice versa. These phase changes are gov-

erned by a minimization of the energy of the

system. The phase transformation is simulated by

allowing the polarization of the crystal to be

switched to any of the possible 14 tetragonal and

rhombohedral directions. Other phase changes such

as AF±F can be implemented by changing the

polarization (phase) switching laws. Such switching

laws should account for the volume change from

the antiferroelectric state to the ferroelectric state

that occurs in materials like PLSnZT.

2.3. Grain orientation eects

Ceramics consist of many randomly orientated

grains. Each grain is modeled as a uniformly polar-

ized cell that has a spontaneous polarization and a

spontaneous strain. Inhomogeneities in the local

electric and stress ®elds are ignored so that each

grain is subjected to the same loading conditions. A

global coordinate system is ®xed in space. A local

coordinate system is assigned to each grain. The

orientation of the local coordinate system relative

to the global coordinate system is de®ned in terms

of nine direction cosines

x j a ji X i 8

where aij cos yij are the direction cosines between

the local and the global coordinate system and the

subscripts refer to Cartesian coordinate directions.

The yij are speci®ed by three randomly generated

Euler angles. The polarization direction of each

grain is de®ned in its local coordinate system. The

unit cell elongates in the direction of polarization

and contracts in the directions perpendicular to itfor both structures.

2.4. Switching laws with intergranular interaction

The switching law for each grain should account

for the dierent energy levels associated with 908

and 1808 switching for the tetragonal structure, or

109.58, 70.58, and 1808 switching for the rhombohe-

dral structure. These energy levels are associated

with the nucleation and propagation energies of

each type of domain wall. The ferroelectric/ferroe-

lastic switching law described for single crystals

must be modi®ed to account for discontinuities in

the spontaneous polarization and strain from grainto grain. These discontinuities are accommodated

by local stress and local electric ®eld. The local

CHEN and LYNCH: SWITCHING OF FERROELECTRIC MATERIALS 5305



stress and electric ®eld increase the energy of the

system. These intergranular interactions can be ap-

proximated as inclusions in a linear piezoelectric

matrix. The problem of the spherical piezoelectricinclusion has been solved by Benveniste [13] and

Huang [14]. Their solutions are too computationally

intensive to be implemented because many thou-

sands of inclusion problems would have to be

solved to simulate the response of a ceramic. In this

work, details of the interactions between each grain

and matrix are not explicitly considered. Rather,

they are approximated by considering the dier-

ences between the spontaneous polarization and

spontaneous strain of the grains and the remanent

polarization and strain of the ceramic. Equation (9)

is used to approximate the interaction energies

between each grain and the ceramic matrixU I U D

I U eI 9

where U DI is the interaction energy due to the mis-

match of electric displacement, and U eI is the inter-

action energy due to the mismatch of strain

between the grain and the matrix. McMeeking and

Hwang [15] have shown that U DI and U e

I can be

approximated by

U DI

c1

2eDs

i À Dri Ds

i À Dri 10

U eI c2

Y

2es

ij À erij es

ij À erij 11

where c1 and c2 are parameters, which would be

chosen so that the computational results are in best

agreement with the experimental data. Dsi , es

ij are

the spontaneous electric displacements and strains

of each grain, respectively. Dri , er

ij are the remanent

electric displacements and strains of the matrix,

which are the average values over all grains.

The work done by the external forces during

polarization switching is given by

W E i DDi sij Deij 12

where

DDi D2i À D1

i 13

Deij e2ij À e

1ij 14

where DDi is the change in the spontaneous polariz-

ation, and Deij the change in spontaneous strain

during switching. Superscript (1) indicates the pre-

sent value, while superscript (2) indicates the new

value if polarization switching occurs. The contri-

bution of elastic strain and electric displacement to

the work and interaction energy are assumed to be

relatively small in comparison with that of spon-

taneous strain and electric displacement. When the

sum of the work done and the reduction of inter-action energy exceed a critical value, polarization

switching occurs. Thus the switching criterion is

de®ned as

W À DU Ir2D0E 0 15

where DU I is the change of interaction energywhich is given as

DU I U 2I À U

1I 16

where D0 is the value of the spontaneous polariz-

ation, E 0 the magnitude of the critical electric ®eld

at which the polarization switching occurs if only

the electric ®eld is applied to a single crystal grain.

Since the energy necessary to move dierent kinds

of domain walls diers, E 0 must be given dierent

values associated with 908 and 1808 switching for

the tetragonal structure, or 70.58, 109.58, and 1808

switching for the rhombohedral structure. E 0 is also

a material dependent parameter.

2.5. Local response to applied loads

The global coordinate system is used to describe

the applied stress and electric ®eld. At each time

step in the computation, components of the applied

stress and electric ®eld are computed in each of the

local coordinate systems. The switching criterion of

equation (15) is checked for every grain and for

every possible polarization direction to see if switch-

ing will occur. If more than one switching direction

can occur, the direction which maximizes the left-

hand side of equation (15) is chosen. If more than

one such direction exists, one of them is chosen ran-

domly. In the computational process, the incrementof loading at each step is limited in size, thus only a

few grains change their polarization directions in

each step. After all possible polarization switches

have occurred, the piezoelectric tensor of each grain

is rotated to the new polarization direction. Next

the strain and electric displacement of each grain

are computed from constitutive equations (1) and

(2) in the local coordinate system.

2.6. Global response to applied loads

The macroscopic response of the ceramic is com-

puted from the volume average response of each

grain. The strain and polarization of each grain arecomputed, and then rotated to the global system

and averaged. The global strain and electric displa-

cement are updated only at the end of each loading

increment. After the global response has been deter-

mined, i.e. the remanent polarization and remanent

strain have been computed, the applied loads are

incremented and the local response is recomputed.

3. RESULTS AND DISCUSSIONS

3.1. Simulations of the behavior of 8/65/35 PLZT

The results of a series of experiments perfomed

on ceramic rhombohedral 8/65/35 PLZT [1] aresimulated. Data include strain±electric ®eld hyster-

esis vs stress, electric displacement±electric ®eld hys-




teresis vs stress, and stress vs strain, and stress vs

electric displacement measured at zero electric ®eld.

Input parameters are chosen so that the compu-

tational results are in agreement with the data.Since 1808 switching does not change the spon-

taneous strain, the sharp tail and negative strain

observed in experiments are due to 70.58 (or 109.58)

switching following by another 70.58 (or 109.58). It

suggests that 70.58 or 109.58 switching is preferred

over 1808 switching in this material. Therefore, the

energy level for 70.58 and 109.58 switching should

be smaller than that for 1808 switching. In the

simulation, E 1800 5E 70X5

0 is used to favor 70.58

switching. The value E 70X50 E 109X5

0 0X1 MVam

is chosen, so that the simulated coercive

®eld is in agreement with the measured one

(E c 0X36 MVam). D0 0X3 Cam2 and e0 0X0039

are used to get remanent polarization

(Dr 0X24 Cam2) and remanent strain (er 0X0014)

at zero stress and electric ®eld. The saturation strain,

esaij 0X0687, gives good simulated results at high

electric ®eld levels. The interaction energy factors,

c1=0.03, c2=0.005, are used to get a sharp shape of

the strain±electric ®eld curve. Piezoelectric coe-cients are selected as d 333 d 113 1X0 Â 10À9 maV to

match the slope of the strain±electric ®eld curve.

3.1.1. Simulated e33 ±E 3 hysteresis loops vs stress.

Initially the material is in the paraelectric state with

zero electric displacement and strain. When an elec-

tric ®eld above the coercive ®eld is applied, the

sample switches to the polarized state. After several

cycles, a stable hysteresis loop develops [Fig. 1(a)].

At zero stress, the simulation results show the sud-

den decrease of strain during switching, but nega-

tive strains are not obtained. At 10 and 30 MPa

compressive stress, the simulations give the same

value of negative strain as the data. At higher stress

levels, the remanent strain steadily decreases and

the curves are shifted downward and become ¯atter.

Elastic deformation and ferroelectric/ferroelastic

Fig. 1. Comparison of simulated and measured longitudinal strain and polarization subjected to anelectric cycle: (a) strain vs electric ®eld at various compressive stress levels; (b) electric displacement vs

electric ®eld at dierent compressive stress levels.




switching contribute to this change. The higher

stress level decreases the coercive ®eld and

encourages the polarization to switch to 70.58 (or

109.58) directions. After the ®rst 70.58 (or 109.58)switching, the stress tends to keep the polarization

in these directions and hinder further switching. A

much higher loading level is necessary to make the

second 70.58 (or 109.58) switch.

3.1.2. Simulated D3 ±E 3 hysteresis loops vs stress.

The simulated electric displacment vs electric ®eld

hysteresis loops are very close to the data at all

stress levels [Fig. 1(b)]. The compressive stress

decreases the remanent polarization. At À10 MPa, a

higher electric ®eld is needed to complete the

switching. At À30 MPa, the electric ®eld is not able

to fully repole the material.

3.1.3. Simulated s3 vs e33 curve. The polarized

ceramic is subjected to uniaxial compressive stress

parallel to the direction of polarization and the

stress±strain curve [Fig. 2(a)] is simulated. The

measured remanent strain at zero stress is smaller

than seen in the simulation. This is because the

model does not include ageing (time dependent

eects), but the material does. Initially, the response

of the ceramic is due to the linear elastic and piezo-

electric eect. When the coercive stress is reached,

the material starts to depole. When the stress has

reached À35 MPa, the switching is nearly complete

and the stress±strain behavior becomes linear elas-

tic. The unloading slope is the same as the loading

slope. When the sample is unloaded to zero, the

®nal strain is negative relative to the unpoled state

due to 70.58 and 109.58 switching.

3.1.4. Simulated s3 vs D3 curve. The stress±elec-

tric displacement curve [Fig. 2(b)] is measuredduring the stress/strain test. The shape of the

stress±electric displacement curve is similar to that

of the stress±strain curve. The initial measured

remanent polarization is smaller than the simulated

one due to ageing. After the maximum amount of

ferroelastic switching has occurred, although the

strain reaches a high negative level, a positive rema-

nent polarization still exists in the ceramic. The

unloading slope has the same sign, but a dierent

value than that of the initial loading slope. When

the specimen is unloaded to zero, the simulated

remanent polarization is smaller than the measured

one.

3.2. Parametric study

A parametric study is performed to investigate

the eects of dierent parameters on material beha-

vior. All of the results are normalized by E 900 (or

E 70X50 for rhombohedral), P0, e0, and s0. Since stress

can drive only 908 (or 70.58, 109.58) switching, s0 is

given by

s0 P0E 90

0

e0

or s0 P0E 70X5

0

e0

X 17

Eects on the material behavior are examined by

varing individual parameters. First, intergranularinteraction eects are studied. Next, the eects of

tetragonal and the rhombohedral structures are

examined, and a morphotropic boundary material

is simulated. Finally, the model is used to validate a

macroscopic law for the evolution of the piezoelec-

tric coecients during switching.

3.2.1. Intergranular interaction eects. Electric dis-

placement vs electric ®eld and strain vs electric ®eld

hysteresis loops are simulated with interaction

(c1=0.05, c2=0.025) and without interaction

(c1=c2=0) (Fig. 3). The interaction between each

single grain and the matrix plays an important role

in the constitutive behavior. The interaction energy

initially increases the coercive electric ®eld by dis-

couraging polarization switching. However, after

the electric displacement and strain change sign, the

interaction energy encourages further switching.

This results in a sharp tail in the strain±electric ®eld

curve at zero stress [Fig. 3(a)]. At zero stress, the

coercive electric ®eld with interaction energy is

about 1.5 times higher than that without. If c1 and

c2 are decreased, these dierences are decreased.

When the stress level is increased, the dierence

becomes less signi®cant. Interaction does not

change the shape, but widens the electric displace-

ment vs electric ®eld loops because more energy

needs to be overcome [Fig. 3(b)]. The switchingsaturates at the same electric ®eld level for these

two cases.

Fig. 2. Comparison of simulated and measured longitudi-nal strain and polarization subjected to a stress cycle: (a)

stress vs strain; (b) stress vs electric displacement curve.




A fully polarized ceramic subjected to a stress

cycle at zero electric ®eld is simulated and stress±

strain and stress±electric displacement curves are

plotted (Fig. 4). The intergranular interaction does

not aect the stress±strain behavior signi®cantly

[Fig. 4(a)], but the coercive ®eld for depolarization

is increased. However, the shapes of stress±electric

displacement curves are changed dramatically by in-

teraction [Fig. 4(b)]. When the compressive stress

reaches a very high level, the remanent polarization

of the ceramic reaches a minimum positive value.

Next, the stress is unloaded to zero. If no inter-

action energy is present, the following tensile stressdoes not repolarize the material because there is

equal probability of the polarization switching to a

positive or negative 908 direction. Since the positive

remanent polarization tends to make the grains

align in one direction to minimize the potential

energy of the ceramic, the tensile stress repolarizes

the ceramic when the stress exceeds the coercive

stress level and the interaction eect is im-

plemented. This gives the stress±strain hysteresis

loop [Fig. 4(b)].

3.2.2. Eects of tetragonal and rhombohedral crys-

tal structures. The same parameters with interaction

are used for tetragonal and rhombohedral ceramicsand strain vs electric ®eld and electric displacement

vs electric ®eld hysteresis loops are generated at

dierent stress levels. The results are very similar

for these two crystal structures. The tails in the

strain vs electric ®eld curves computed from the

rhombohedral structure are a little shorter thanthose from the tetragonal at zero stress [Fig. 5(a)].

The coercive ®eld of the rhombohedral structure is

a little smaller than the tetragonal. When the stress

is increased, the dierences become smaller. The

electric displacement vs electric ®eld curves at dier-

ent stress levels are almost the same for these two

ceramics [Fig. 5(b)]. The curves obtained from these

two structures are so close that we may conclude

that the dierences between the crystal structures

do not have much in¯uence on the nonlinear beha-

vior or the hysteresis of ferroelectric materials.

3.2.3. Eect of phase changes. A morphotropic

boundary composition is simulated by allowing

switching into either tetragonal or rhombohedral

sites. The results show the tails disappear when

both tetragonal and rhombohedral sites are avail-

able [Fig. 6(a)]. The remanent strain at zero electric

®eld is higher for the mixed phase, since more

polarization directions (all 14 possible tetragonal

and rhombohedral directions) are possible, and the

polarization can be aligned more closely with a

given direction. Similarly, the remanent polarization

of the mixed phase material is higher than tetra-

gonal or rhombohedral alone [Fig. 6(b)].

3.2.4. Check of phenomenological law. The consti-

tutive law is applied to every grain in the local

coordinate system so that the response of each

Fig. 3. Comparison of simulated results with or withoutinteraction energy eect at dierent zero compressivestress level and twice the coercive stress level: (a) strain vs

electric ®eld; (b) electric displacement vs electric ®eld.

Fig. 4. Comparison of simulated results at zero electric®eld with or without interaction energy eect: (a) stress vs

strain; (b) stress vs electric displacement.




grain can be evaluated. This makes it possible to

use anisotropic material properties and track their

change during switching. Since the macroscopic re-

sponse of the ceramic is completely determined by

the properties of grains, this model is very useful in

testing macroscopic phenomenological constitutive

laws that include switching eects. A macroscopic

equation used for the piezoelectricity tensor is

tested. Initially the unpoled ceramic is macroscopi-cally isotropic and has no piezoelectric properties

(piezoelectricity tensor is zero). When the material

is poled by a large electric ®eld, it shows a strong

piezoelectric eect. The value of the piezoelectricity

tensor is related to the remanent polarization. An

approximate relationship between the piezoelectri-city tensor and the remanent polarization is that the

value of the piezoelectricity tensor is proportional

to the remanent polarization [1], i.e.

d ijk d ijkDr

Drs

18

where d ijk is the piezoelectric coecient of the fully

poled polycrystalline ceramic, Dr the current value

of the remanent polarization, and Drs the fully poled

saturation magnitude of the remanent polarization

of the ceramic. Several simulated compressive stres-

ses are applied to fully poled ceramic to obtain

dierent remanent polarizations. An electric ®eld isthen applied and d 333 is obtained. An approximately

linear relationship between d 333 and Dr is found at

dierent polarization levels (Fig. 7). The approxi-

mate equations for other components of the piezo-

electricity tensor can be checked in the same way. It

appears that equation (18) is a reasonable approxi-

mation for the piezoelectricity tensor during switch-

ing.

4. CONCLUDING REMARKS

A computational model based on micromechanics

has been developed to simulate the nonlinear beha-

vior of ferroelectric ceramics. The ceramics were

Fig. 5. Comparison of simulated results for tetragonal andrhombohedral structures: (a) strain vs electric ®eld; (b)

electric displacement vs electric ®eld.

Fig. 6. Eect of phase changes driven by electric ®eld: (a)strain vs electric ®eld; (b) electric displacement vs electric

®eld.




considered as an aggregate of many grains which

were modeled as single domain with switching. A

linear saturation model was applied to each unit

cell. The strain and electric displacement of eachgrain were obtained by computing the linear com-

ponents and spontaneous components in a local

coordinate system. The macroscopic response was

obtained by averaging over all grains. A switching

criterion that accounts for the interaction between

grains and the dierent energy levels necessary to

move 908 (70.58, 109.58) and 1808 domain walls was

implemented in this model. When 908, 70.58 switch-

ing is favored, the simulation results better ®t the

rhombohedral experimental data. This implies that

it is easier to move 908, 70.58 domain walls than

1808 domain walls. A parametric study was per-

formed using this model. The results suggest the in-

teraction energy has in¯uence on the hysteresis

loops of ferroelectric materials, especially under

mechanical loading. The crystal structures, tetra-

gonal or rhombohedral, do not have a signi®cantly

dierent eect on the nonlinearities in material

behavior. Morphotropic boundary materials were

simulated. They show higher remanent strain and

polarization. This model is capable of predicting

the behavior of ferroelectric materials under multi-

axial loading (mechanical and/or electrical). The

results give us important information that will be

used to guide the development of multiaxial, non-

linear, hysteretic phenomenological constitutive law

for ferroelectric materials.

Acknowledgements ÐWe gratefully acknowledge fundingfor this work by an oce of Naval Research YoungInvestigator award #N0014-96-1-0711 and by a NationalScience Foundation CAREER award, NSF #CMS-9702169.

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Fig. 7. d 333 vs remanent polarization (Dr) curve.Approximate linear relationship is found.


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A Micro Electro Mechanical Model for Polarization Swithing of Ferroelectric Materials

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