a constitutive model for thermo-electro-mechanical behavior …nonlinear behavior of ferroelectric...

International Journal of Solids and Structures 48 (2011) 1318–1329

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International Journal of Solids and Structures

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A constitutive model for thermo-electro-mechanical behavior of ferroelectricpolycrystals near room temperature

Sang-Joo Kim ⇑Department of Mechanical and Information Engineering, University of Seoul, Dongdaemun-gu, Seoul 130-743, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 August 2010Received in revised form 16 December 2010Available online 26 January 2011

Keywords:Constitutive modelPolycrystalThermo-electro-mechanicalFree energyFerroelectricTemperature

0020-7683/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2011.01.018

⇑ Tel.: +82 2 2210 2757; fax: +82 2 2210 5575.E-mail address: [email protected]

A constitutive model that can be used to predict thermo-electro-mechanical linear and nonlinear behav-ior of ferroelectric polycrystals near room temperature is proposed. A ferroelectric polycrystal is modeledby an agglomerate of 210 single crystallites that are distributed regularly over all directions. A variant in asingle crystallite is characterized by a Gibbs free energy function whose coefficients have linear depen-dency on temperature. A dissipation inequality for domain switching is derived from the restriction ofthe second law of thermodynamics. Domain switching process is governed by a viscoplastic switchinglaw with temperature-dependent switching parameters. The responses of the proposed model to electricfield and mechanical stress loading at room and elevated temperatures are calculated and comparedqualitatively with experimental observations available in literature.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

A ferroelectric polycrystal such as lead zirconate titanate (PZT)is composed of a number of single crystallites whose orientationsare distributed over all directions and each constituent singlecrystallite makes a complicated mesoscopic domain structure con-sisting of six distinct types of variants at room and elevated tem-peratures. A domain is a homogeneous region within which onlya single type of variant exists and it is separated from neighboringdomains of distinct types by domain boundaries. As external stim-uli are applied in the form of stress, electric field or temperaturechange, a domain boundary between two distinct types of variantsmoves in the way that the more favorable type of variant isincreased and the other is decreased, which is called domainswitching. Mesoscopic domain switching is often used to explainmacroscopic nonlinear behavior of ferroelectric materials such aselectric displacement hysteresis loops, butterfly strain curves,creep, and so on. All these nonlinear behavior must be understoodand modeled properly for wide and efficient applications of theinteresting materials.

Most of recent constitutive models for ferroelectric polycrystalsare proposed for room temperature behavior of the materials. Forexample, Huber et al. (1999) proposed a constitutive model forferroelectric polycrystals and Huber and Fleck (2001) discussed

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about multi-axial electrical switching of a ferroelectric materialboth experimentally and theoretically. Landis (2002) constructeda general form for multi-axial constitutive laws for ferroelectricceramics, which is founded on an assumed form of a Helmholtzfree energy, postulated switching surfaces, and associated flowrules. Kamlah et al. (2005) used a multidomain single crystalswitching model to calculate the poling behavior of ferroelectricpolycrystals. Kim and Jiang (2002) proposed a finite element modelfor rate-dependent behavior of ferroelectric ceramics; Kim (2007a)predicted polycrystalline behavior of ferroelectric materials using arepresentative volume element model obtained by combining theregular dodecahedron model of Huber and Fleck (2001) and the cu-bic model of Belov and Kreher (2005), Kim (2009) predicted thetensile creep behavior of a PZT wafer by introducing a normallydistributed free energy model. Li and Fang (2004) predicted do-main switching behavior in ferroelectric materials by a three-dimensional finite element model; and Li et al. (2007) proposed asimple constitutive model based on an analysis of physical mech-anisms of domain switching and calculated the responses of thematerials to uncoupled electro-mechanical loading. Pathak andMcMeeking (2008) proposed a three-dimensional finite elementmethod based on the constitutive model of Huber et al., 1999)and calculated the nonlinear behavior of polycrystalline ferroelec-tric materials under electro-mechanical loadings. Klinkel (2006)developed a macroscopic constitutive law that is thermodynami-cally consistent and that is determined by two scalar functions:the Helmholtz free energy and a switching surface.

However, all of these models for ferroelectric ceramics are onlyfor room temperature behavior of the materials. In the applications

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S.-J. Kim / International Journal of Solids and Structures 48 (2011) 1318–1329 1319

of ferroelectric ceramics such as actuators and sensors, the materi-als are usually subjected to a wide range of temperature changedue to either the change in ambient temperature or a high temper-ature generated by the self-heating of the dissipative materials. Forexample, a temperature increase more than 75 �C was observedduring an operation of a piezoelectric stack actuator at 130 Hzdriving frequency at a surrounding temperature of 120 �C, whichis shown to deteriorate the function of the actuator by depolariza-tion, see Sakai and Kawamoto (1998). It is, therefore, necessary andimportant to develop a constitutive model that can be used to pre-dict nonlinear high temperature behavior of the materials for reli-able design of piezoelectric devices. Recently, Su and Weng (2005)have proposed a Helmholtz free energy potential including thermalenergy terms for a two-phase crystal consisting of paraelectric andferroelectric phases and studied the shift of Curie temperature inthe presence of mechanical stress and electric field. Smith andHom (2005) proposed a temperature-dependent constitutive mod-el for relaxor ferroelectrics by assuming the material is comprisedof an aggregate of micropolar regions having a range of Curie tem-perature. Their model, however, has a restriction that it can simu-late only electrical hysteresis loops of the material. Kim (2007b)generalized the three-dimensional electro-mechanical free energypotential of Kim and Seelecke (2007) so that it may model variousthermal phenomena of perovskite type materials. In Kim’s model,linear moduli of a tetragonal variant are assumed to be constantand independent of temperature. This constant linear moduliassumption during temperature change is acceptable only for afew material properties like pyroelectric and thermal expansioncoefficients, as was verified experimentally by Kim et al. (2010),Kim and Kim (2010). The experimental work of Webber et al.(2009) also shows that elastic compliance remains approximatelyconstant for a significant change of temperature. However, thereare many other linear moduli that can not be considered to be con-stant over temperature. For example, permittivity and piezoelec-tric coefficient are shown to vary about 50% in a temperaturechange of about 100 �C in Kim et al. (2010). Considering that thetemperature-dependent linear moduli may have influence on thenonlinear behavior of ferroelectric ceramics as well as its linearbehavior, the constant linear moduli assumption has to beimproved.

In the present study, the constitutive model for thermo-electro-mechanical behavior of ferroelectric ceramics developed by Kim(2007b) is improved so that the variation of linear moduli andother material parameters over temperature may be included inthe model. It is assumed that linear moduli and material parame-ters have linear dependency on temperature and thus each of themcan be expressed as a linear function of temperature. Then eachdistinct type of tetragonal variants is characterized by a Gibbs freeenergy function with temperature-dependent linear moduli. TheGibbs free energy function is derived from thermodynamic defini-tions of material linear moduli. A ferroelectric polycrystal ismodeled by a mixture of 210 single crystallites each of which isregularly distributed over all directions, as proposed by Kim(2007b). A thermodynamic theoretical framework for a ferroelec-tric polycrystal is proposed and a dissipation inequality consistentwith the second law of thermodynamics is derived. The phenome-nological viscoplastic law for domain switching proposed by Huberand Fleck (2001) is modified so that the switching parameters suchas critical driving force and switching rate are expressed as linearfunctions of temperature. The macroscopic behavior of a ferroelec-tric polycrystal is obtained by averaging the behavior of each con-stituent variant calculated from Gibbs free energy function andswitching evolution equation. The calculated responses of ferro-electric polycrystals to electric field and mechanical stress at roomand elevated temperatures are discussed and compared qualita-tively with experimental observations.

2. Thermodynamic framework

A ferroelectric polycrystal is composed of a sufficient number ofsingle crystallites whose crystallographic orientations are distrib-uted uniformly over all directions. Each crystallite (or grain) inthe polycrystal has its own unique crystallographic orientationand has a complicated domain structure. A domain in a crystalliteis a homogeneous region within which strain and polarization areconstant. Assuming negligible contributions of domain walls orgrain boundaries, the macroscopic behavior of a ferroelectric poly-crystal can be calculated from those of constituent variants in thepolycrystal. If a ferroelectric polycrystal is composed of Ncrystal-lites and a crystallite has m different types of variants, then thepolycrystal has a total of (N � m) variants. In the present paper, itis assumed that a a-type variant in the nth crystallite is character-ized by Helmholtz free energy per unit mass wna(Dna,Sna,h), whichis a function of electric displacement vector Dna, strain tensor Sna,and absolute temperature h. In an isothermal thermodynamic pro-cess with uniform temperature, the Clausius–Duhem inequalityhas the form:

�q _wna � qgna _hþ E � _Dna þ T � _Sna P 0; ð1Þ

for a thermoelastic polarizable body of the a-type variant in the nthcrystallite, where qis density, E electric field intensity vector, Tstress tensor, and gna entropy per unit mass of the a-type variant.Enforcing the dissipation inequality (1) yields the potentialrelations:

E ¼ q ownaoDna

Dna; Sna; hð Þ; ð2Þ

T ¼ q ownaoSna


gna ¼ �ownaoh

Dna; Sna; hð Þ ð4Þ

as necessary and sufficient conditions for the dissipation inequalityto be satisfied for all thermo-electro-mechanical processes. Assum-ing (2) and (3) are solvable for D na and Sna and introducing Gibbsfree energy per unit volume gna by the Legendre transformation:

gna ¼ qwna � E � Dna � T � Sna; ð5Þ

then we get the relations:

Dna ¼ �ognaoE

E;T; hð Þ; ð6Þ

Sna ¼ �ognaoT

E;T; hð Þ; ð7Þ

gna ¼ �1q

ognaoh

E;T; hð Þ: ð8Þ

Next turn to a ferroelectric polycrystal consisting of N single crystal-lites. A couple of assumptions are made to simplify the analysis ofthe polycrystal. First, taking the Reuss-type assumption, the stressT, the electric field E, and the temperature hare uniform in the poly-crystal. Thus all variants in the polycrystal are subjected to the sameelectric, stress, and temperature fields. The second assumption isthat the contributions of domain walls or grain boundaries to thewhole energy of the body are negligible compared to those ofconstituent variants. Then the Helmholtz free energy per unit massof the polycrystal wcan be obtained by adding the correspondingenergies of constituent variants of the polycrystal:

w ¼ 1N

XNn¼1

Xma¼1

nnawna Dna; Sna; hð Þ; ð9Þ

where nna(>0) and wna are the volume fraction and the Helmholtzfree energy per unit mass of a-type variant in the nth crystallite,

1320 S.-J. Kim / International Journal of Solids and Structures 48 (2011) 1318–1329

respectively. Since domain switching occurs between variantswithin a crystallite only, the volume fraction nna should satisfy foreach crystallite:Xma¼1

nna ¼ 1: ð10Þ

Finally, the electric displacement D and the strain S of the polycrys-tal are assumed to be the sums of those of constituent variants:

D ¼ 1N

XNn¼1

Xma¼1

nnaDna; ð11Þ

S ¼ 1N

XNn¼1

Xma¼1

nnaSna: ð12Þ

For a thermoelastic polarizable body characterized by the Helm-holtz free energy (9), the Clausius–Duhem inequality is expressedas:

�q _w� qg _hþ E � _Dþ T � _S P 0; ð13Þ

in an isothermal thermodynamic process with uniform tempera-ture. Inserting (9), (11) and (12) into (13) and using (4) and (5)yields the following relations:

E ¼ q ownaoDna


T ¼ q ownaoSna


g ¼ 1N

XNn¼1

Xma¼1

nnagna; ð16Þ

�Xma¼1

gna _nna P 0: ð17Þ

Introducing the thermodynamic driving force defined by

fnab ¼ gna � gnb; ð18Þ

the inequality (17) can be reduced either to:Xma¼1

Xmb¼1

fnab _nnab P 0; ð19Þ

for each crystallite or to:

fnab _nnab P 0; ð20Þ

for a specific type of switching, where _nnab is the rate of switchingfrom variant ato variant bin the nth crystallite and _nnaa ¼ 0 withno summation on a.

3. Thermo-electro-mechanical constitutive model

Linear moduli of ferroelectric polycrystals can be measuredunder various different conditions. They have different valuesdepending on the conditions at which they are measured. Forexample, elastic compliance coefficients can be measured undereither isothermal or adiabatic conditions, and their values obtainedfrom measurements in one condition are related to those in othercondition according to thermodynamic relations. In case of ferro-electric materials which are coupled thermo-electro-mechanically,it is more convenient to measure linear moduli under the condi-tions of constant electric field, stress and temperature fields. Forexample, elastic compliance coefficients are measured with moreeasy at the condition of constant electric field, stress, and temper-ature than, for example, at the condition of constant electric dis-placement, strain, and entropy. If one wants to derive a freeenergy potential using a set of material properties measured atconstant electric field, stress and temperature conditions, then it

is more convenient to construct a Gibbs free energy potential thanother forms of free energy. In this point of view, a specific form ofGibbs free energy function for ferroelectric variants is derivedbased on linear moduli measured at the conditions of constantelectric field, stress, and temperature.

The values of most linear moduli change depending on temper-ature. Their dependence on temperature would be nonlinear if therange of temperature considered is relatively large. However, re-stricted within a relatively small range of temperature, then theycan be considered to either change linearly with temperature or re-main constant. For example, the values of piezoelectric coefficientsand permittivity of PZT-5H are observed to change linearly withtemperature in the temperature range between � 80 and 80 �Cby Wang et al. (1998); similar linear temperature dependence ofpiezoelectric coefficient and permittivity was also observed in aPZT 3203HD wafer between 20 and 110 �C by Kim et al. (2010);in addition, Kim and Kim (2010) has shown that the values of ther-mal expansion and pyroelectric coefficients of a PZT 3203HD waferremain approximately constant in the temperature range between20 and 110 �C. It has long been considered by the community offerroelectric materials scientists that the temperature dependenceof permittivity is governed by the Curie–Weiss law for the temper-ature range near and above the Curie point. However, the Curie–Weiss law is not taken here for simulation, because the tempera-ture range of present interest is near room temperature and thusit is far below the Curie point. Based on these observations, the fol-lowing relation is assumed for a typical linear modulus or materialparameter A.

bAðhÞ ¼ Ahðh� h0Þ þ A0; ð21Þwhere Ah represents a constant rate of change in linear modulus Ao-ver temperature and A0 is the value of A at the reference tempera-ture h0. The subscript hin Ah means that it is the rate of change overtemperature. The linear modulus represented by A can be either ascalar or a vector or a tensor.

Now we discuss the derivation of Gibbs free energy potential. Ascan be found in Nye (1985), Gibbs free energy per unit volume for a-type variant in the nth crystallite ĝnaðE; T; hÞ is differentially re-lated to various linear moduli by

� o2ĝnaoE2

!T;h

¼ ĵT;hna ðhÞ; �o2ĝnaoToE

!h

¼ d̂hnaðhÞ; ð22Þ

� o2ĝnaoT2

!E;h

¼ ŝE;hna ðhÞ; �o2ĝnaohoT

!E

¼ âE0naðhÞ; ð23Þ

� 1q

o2ĝnaoh2

!E;T

¼ ĉET0ðhÞ

h; � o

2ĝnaohoE

!T

¼ p̂T0naðhÞ; ð24Þ

where j, s, c, d, a and p represent permittivity tensor, elastic com-pliance tensor, specific heat, piezoelectric tensor, thermal expansiontensor, and pyroelectric vector, respectively. The superscripts E,T orhon a linear modulus mean that the linear modulus is measured un-der constant electric field, stress, or temperature, respectively. Thesuperscript 0 after E, T, or ET in (23)2 and (24) means that the linearmoduli are measured at the condition of zero stress and electricfields. For example, aE0 in (23)2 is measured at zero electric andstress fields. This restriction of measurement condition is neededto guarantee thermodynamical consistency between various mate-rial moduli in (22)–(24) that are assumed to depend linearly ontemperature. Since the value of specific heat c is taken to be thesame for all variants in the polycrystal, the subscript na is omittedin (24)1. Integrating the above six relations and imposing appropri-ate conditions at h = h0, the Gibbs free energy per unit volume


ĝnaðE;T; hÞ for a-type variant in the nth crystallite can be proposedas:

ĝnaðE; T; hÞ ¼ � ð1=2ÞĵT;hna ðhÞE � E� ð1=2ÞŝE;hna ðhÞT � T� d̂hnaðhÞT � Eþ q cET00 � cET0h h0

� �ðh� h0Þ � h lnðh=h0Þ½ �

� ð1=2ÞqcET0h ðh� h0Þ � E � bDRnaðhÞ � T � bSRnaðhÞ; ð25Þwhere bDRnaðhÞ and bSRnaðhÞ represent the remanent polarization andthe remanent strain at temperature hand they are given by

bDRnaðhÞ ¼ DR0na þ ð1=2Þ p̂T0naðhÞ þ p̂T0naðh0Þ� �ðh� h0Þ;bSRnaðhÞ ¼ SR0na þ ð1=2Þ âE0naðhÞ þ âE0naðh0Þ� �ðh� h0Þ: ð26ÞAs can be seen in (26), DR0na and S

R0na are remanent electric displace-

ment and remanent strain at the reference temperature h = h0 andthey are referred to as reference remanent polarization and refer-ence remanent strain, respectively. Electric displacement Dna, strainSna, and specific entropy gna of a-type variant in the nth crystallitecan be calculated by inserting (25) and (26) into (6)–(8):

bDnaðE;T; hÞ ¼ ĵT;hna ðhÞEþ d̂hnaðhÞTþ bDRnaðhÞ;bSnaðE;T; hÞ ¼ ŝE;hna ðhÞTþ d̂hTnaðhÞEþ bSRnaðhÞ;qĝnaðE;T; hÞ ¼ p̂T0naðhÞ � Eþ âE0naðhÞ � Tþ qcET0h ðh� h0Þ

þ q cET00 � cET0h h0� �

lnðh=h0Þþ ð1=2ÞjT;hhnaE � Eþ ð1=2Þs

E;hhnaT � Tþ d

hhnaT � E:

ð27Þ

Differentiating bDna in (27)1 with respect to temperature, one canobtain a general expression of pyroelectric coefficient that dependson stress and electric fields as well as temperature. It is found to begiven by p̂TnaðE;T; hÞ ¼ ðoĵT;hna =ohÞEþ ðod̂hna=ohÞTþ p̂T0na, where ĵT;hna ; d̂hnaand p̂T0na are assumed to depend linearly on temperature in (21).Similarly, general expressions for thermal expansion coefficientand specific heat can be obtained by differentiating (27)2 and(27)3 with respect to temperature, respectively.

Using (26) and (27), the Gibbs energy gna in (25) can be ex-pressed as:

gna ¼ �ð1=2ÞE � Dna þDRna

� �� ð1=2ÞT � Sna þ SRna

� �þ qðcET00

� cET0h h0Þ h� h0ð Þ � h lnðh=h0Þ½ � � ð1=2ÞqcET0h h� h0ð Þ2; ð28Þ

where Dna and Sna are given by (27) and DRna and S

Rna are given by

(26). Inserting (25) into (18) yields the thermodynamic drivingforce fnab in terms of E,T and h:

f̂ nabðE;T; hÞ ¼ �ð1=2ÞDĵT;hnabðhÞE � E� ð1=2ÞDŝE;hnabðhÞT � T

� Dd̂hnabðhÞT � E� E � DbDRnabðhÞ � T � DbSRnabðhÞ; ð29Þwhich can be reduced to the driving force expression of Kim(2007b) by assuming constant linear moduli in the temperaturerange of interest. The driving force fnab in (29) can also be consid-ered as a generalized version of the driving force expression at aconstant temperature given by Huber et al. (1999), Kim and See-lecke (2007). Assuming that the degree of anisotropy in elasticand dielectric moduli is relatively small compared to that of piezo-electric moduli, the expression for driving force (29) is reduced to:

f̂ nabðE;T; hÞ ¼ �E � DbDRnabðhÞ � T � DbSRnabðhÞ � Dd̂hnabðhÞT � E; ð30Þwhich is a generalized version of the driving force expression at aconstant temperature given by Huber and Fleck (2001). The expres-sion knab = h(gna � gnb), called latent heat, represents the heat gen-erated when a unit mass of material moves from variant ato variantbunder a condition of zero thermodynamic driving force. The latent

heats for various types of switchings are calculated from (27)3 andgiven by

qk̂nabðE;T; hÞ ¼ h Dp̂T0nabðhÞ � Eþ DâE0nabðhÞ � Tþ12

DjT;hhnabE � E�þ 1

2DsE;hhnabT � T þ Dd

hhnabT � E

: ð31Þ

In the equations from (29)–(31), DAnab = Ana � Anb with A either ascalar or a vector or a tensor.

Turning to the evolution law of domain switching, the rate ofswitching _nnab from variant ato variant bin the nth crystallite is of-ten proposed as a function of thermodynamic driving force in arate-dependent model of ferroelectric materials, for example, seeHuber and Fleck (2001), Kamlah et al. (2005), Kim and Jiang(2002), Pathak and McMeeking (2008), and so on. All these modelsfor domain switching are proposed to simulate the electro-mechanical responses of the materials at constant room tempera-ture, though a simple evolution law with a temperature-dependentswitching coefficient was proposed by Kim (2009). In order to sim-ulate the thermo-electro-mechanical responses of the materials, itis necessary to understand the effects of temperature on the do-main switching of the materials. The recent works of Kim et al.(2010), Kim (2010) show that the magnitude of coercive field de-creases with increase in temperature, which is also observed byKounga et al. (2008), Mitoseriu et al. (1996), and many others. Inaddition, by introducing the so-called reference remanent polariza-tion and reference remanent strain, they proposed that the rate ofdomain switching becomes faster at higher temperatures. Theseobservations on the effects of temperature on domain switchingprocesses should be reflected in the generalization of previous evo-lution laws. In the present calculations, the viscoplastic law pro-posed by Huber and Fleck (2001) and modified by Pathak andMcMeeking (2008) is taken and generalized. Here the rate of do-main switching _nnab from variant ato variant bin the nth crystalliteis given by

_nnab ¼ _n0fnab

f̂ cðhÞ

m̂ðhÞ�1fnab

f̂ cðhÞ; ð32Þ

where m̂ðhÞ is a creep exponent at temperature h and _n0 is the rateof switching at fnab = fc(h). _n0 is assumed to be constant for the tem-perature range of interest. With m̂ðhÞ � 1; f̂ cðhÞ becomes the criticalvalue of fnab at which there is a rapid increase in switching rate fromvariant a to variant b in the nth crystallite at temperature h. Thecreep exponent m̂ðhÞ controls the shape of the hysteresis responsesat the onset of ferroelectric switching at temperature h, high valuesof m̂ðhÞ corresponding to a sudden onset of switching. It can be eas-ily seen that the proposed evolution law (32) is reduced to that ofPathak and McMeeking (2008) at the reference temperature h0.

The evolution law (32) can be used to calculate the responses ofa single crystallite with a particular set of switching systems. Forexample, consider a ferroelectric crystallite composed of six dis-tinct types of tetragonal lattices. The average electric displacementDn and the average strain Sn of the crystallite can be obtained from:

Dn ¼X6a¼1

nnaDna; Sn ¼X6a¼1

nnaSna; ð33Þ

where the electric displacement Dna and the strain Sna of a-typevariant in the nth crystallite are calculated by (27). Paying attentionto the macroscopic responses of a ferroelectric polycrystal, it is nec-essary to introduce a set of crystallite systems which would repro-duce the isotropic response of an unpoled ferroelectric polycrystal.In the present paper, the representative volume element model ofKim (2007b), which is introduced by combining the dodecahedron


model of Huber and Fleck (2001) and the cube model of Belov andKreher (2005), is used to predict the macroscopic behavior offerroelectric polycrystals. Kim’s RVE model consists of N singlecrystallites whose orientations are distributed uniformly over alldirections. The orientation of each constituent crystallite is deter-mined when the orientation of the frame of the crystallite isdescribed with respect to a fixed global frame. The origin of the glo-bal frame (x1,x2,x3) is at the center of the regular dodecahedronshown in Fig. 1(a). The global frame is located so that the x3 axispasses through one of the twelve regular pentagon faces. There ex-ist 10 vertices in the upper half of the dodecahedron, which arenumbered from 1 to 10, as shown in Fig. 1(a). Now an intermediateframe (y1,y2,y3) is introduced. The y3 axis of the intermediate framestarts from the origin of the global frame and passes through one ofthe vertices. The y3 axis of the intermediate frame shown in Fig. 1(a)passes through the vertex point 2. There are three straight linesconnecting the vertex point 2 and three neighboring vertices 1, 7,and 3. The y1 axis of the intermediate frame is chosen to passthrough the vertex point 7 in the figure. The y2 axis of the frameis automatically determined from the directions of the y1 and y3axes. Two more intermediate frames can be introduced with thesame y3 axis: the y1 axis in one of the frames passes through vertexpoint 1 and it passes through vertex point 7 in the other. Thus, threeintermediate frames are introduced associated with the vertex

Fig. 1. Orientations of constituent crystallite frames in the

point 2. Since there are ten different vertices in the upper half ofthe dodecahedron, a total of 30 intermediate frames can beintroduced.

Now a local frame (z1,z2,z3) is introduced whose orientation isdescribed with respect to an intermediate frame (y1,y2,y3). A cubeis drawn in Fig. 1(b) so that three orthogonal sides of the cube coin-cide with the three axes of the intermediate frame (y1,y2,y3). Thereexist six diagonal planes in the cube and one of them is shown inFig. 1(b). A local frame (z1,z2,z3) is defined on the diagonal plane.The z2 axis of the local frame is perpendicular to the diagonal planeand the other two z1 and z3 axes are on the diagonal plane, asshown in the figure. Since there are six diagonal planes in the cube,six different local frames can be introduced associated with thecube associated with the intermediate frame. If including theintermediate frame itself, then seven independent frames can beintroduced associated with the intermediate frame (y1,y2,y3).Considering thirty intermediate frames introduced in Fig. 1(a), atotal of 210 independent frames can be introduced. A more conciserepresentative volume element can be obtained by neglecting alllocal frames in Fig. 1(b). In this case, the representative volume ele-ment model consists of 30 intermediate frames only. If the numberof single crystallites in the representative volume element is in-creased, then the results would be more accurate. However, sincethe amount of computation is also increased, it is necessary to

representative volume element model of Kim (2007b).


choose a proper number of crystallites in the volume element. Formore details, one may refer to Kim (2007b).

Using the proposed representative volume element model for aferroelectric polycrystal, the macroscopic electric displacement Dand the macroscopic strain S of the polycrystal are given by

D ¼ 1N

XNn¼1

Dn; S ¼1N

XNn¼1

Sn; ð34Þ

where Dn and Sn are the average electric displacement and the aver-age strain of the nth crystallite in the polycrystal given by (33). Thevalue of N can be either 30 or 210, depending on whether the localframes (z1,z2,z3) are included or not in the representative volumeelement model.

4. Results and discussion

In this section, the responses of the model described in previoussections are calculated for several different cases of electric andmechanical loading at room and elevated temperatures. First, elec-tric displacement-electric field curves (or polarization hysteresisloops) and strain-electric field curves (or butterfly strain loops) ofa PZT wafer under through-thickness electric fields at various tem-peratures between 20 and 120 �C are calculated and comparedqualitatively with experimental observations reported in litera-ture. Then mechanical depolarizations by compressive stressesapplied in the direction of poling at various temperatures are cal-culated and they are compared qualitatively with the experimentalmeasurements of Webber et al. (2009). For the calculations, thematerial properties of a ferroelectric tetragonal lattice have to begiven for the temperature range between 20 and 120 �C. Some ofthe material properties are obtained from the experimental obser-vations of Kim et al. (2010), Kim and Kim (2010) on a poled PZTwafer; some are provided by manufacturers; lastly, some are ob-tained from literature. In real materials, the values of materialproperties usually depend on temperature. However, it is also re-ported that the changes in the values of some material propertiesover a relatively small temperature range are either vanished ornegligibly small compared to others. For example, the changes inthe values of elastic stiffness are significantly smaller than thosein permittivity or piezoelectric coefficient during temperaturechange, as discussed by Jaffe and Cook (1971), Wang et al.(1998). In the present calculations, the materials are assumed tobe isotropic mechanically and the values of elastic compliancesE,h and Poisson’s ratio mE,h at constant electric field and tempera-ture are taken to be constant over the temperature range of inter-est. Their values are given by manufacturers (CTS Wireless, USA,http://www.ctscorp.com) for a PZT 3203HD wafer in (35). Pyro-electric and thermal expansion coefficients at zero stress and elec-tric fields are also taken to be constant over a temperature change,which was verified experimentally by Kim et al. (2010), Kim andKim (2010) on a PZT wafer. The value of pyroelectric coefficientpT0c in (35) was obtained from Kim et al. (2010), Kim and Kim(2010). The subscripts c and a, often expressed by 3 and 1 respec-tively, represent the crystallographic directions of a ferroelectrictetragonal lattice in the parallel and perpendicular directions tothe dipole moment of the lattice, respectively. The value of thermalexpansion coefficient aE0a in the direction perpendicular to the pol-ing direction for a 3203HD PZT wafer can be estimated from themeasurements of Kim et al. (2010), where it was measured thataE0a ¼ 4:17� 10

�6=�C over the temperature range of interest. The

value of aE0c for the same PZT wafer is not available at this momentof time. In case of a PIC 255 PZT ceramic in the PI Ceramic (http://www.parameter.se/products/pgfiles/5423de2b-54b5-4427-b4d5-a9b5dcb885d0.pdf), it was observed that, approximately, aE0c ¼�ð1=2ÞaE0a at room and elevated temperatures. Assuming the same

relation holds for a PZT wafer, the value of thermal expansion coef-ficient in the poling direction is aE0c ¼ �2:09� 10

�6=�C. The specific

heat at zero electric field and stress cET0 is also taken to be constantin the temperature range studied, whose value in (35) was pro-vided by manufacturers.

sE;h ¼ 18:8� 10�12 m2=N; mE;h ¼ 0:31; aE0c ¼ �2:09� 10�6=

�C;

pT0c ¼ �6:8� 10�4 C=ðm2 �CÞ; pT0a ¼ 0; aE0a ¼ 4:17� 10

�6=�C;

cET0 ¼ 530 J=ð�C kgÞ:ð35Þ

Now we turn to the material properties whose values cannot be as-sumed constant for temperature change. Permittivity at constantstress and temperature jT,h is assumed to be isotropic and to de-pend linearly on temperature. The linear dependency of permittiv-ity on temperature was observed experimentally by Wang et al.(1998) on a PZT-5H ceramic and by Kim et al. (2010) on a3203HD PZT wafer. The values of aj and jT;h0 , where aj is the rateof change in jT,h over temperature and jT;h0 is the value of j

T,h atthe reference temperature h 0, are measured by Kim et al. (2010)and given in (37). It was also shown experimentally by Wanget al. (1998), Kim et al. (2010) that the piezoelectric coefficients de-pend linearly on temperature in a relatively wide range of temper-ature. The linear temperature dependency of dh31 is also expressedby ad31 and d

h310 and their values measured for a PZT wafer by

Kim et al. (2010) are given in (37). The piezoelectric coefficientdh33 is determined from d

h33 þ 2d

h31 ¼ 0 using the constant volume

assumption under electric field E3, as discussed by Wang et al.(1998). It is shown in Fig. 1 of Wang et al. (1998) that the valueof dh15 is about 1.67 times as great as that of d

h33 and the rate of

change in dh15 over temperature is almost four times as great as thatof dh33 for a PZT-5H ceramic. The values of d

h33 and d

h15 for a 3203HD

PZT wafer can be obtained by using the observations of Wang et al.(1998). Summarizing, the three piezoelectric coefficients for a PZTwafer are assumed to be given byd̂h31ðhÞ ¼ ad31ðh� h0Þ þ d

h310;

d̂h33ðhÞ ¼ �2ad31ðh� h0Þ � 2dh310;

d̂h15ðhÞ ¼ �8ad31ðh� h0Þ � 3:33dh310;

ð36Þ

and the values of material parameters associated with permittivityand piezoelectric coefficients are:

jh0 ¼ 4:51� 10�8 C=ðV mÞ; aj ¼ 2:14� 10�10 C=ðV m �CÞ;

dh310 ¼ �4:11� 10�10 m=V; ad31 ¼ �1:29� 10�12 m=ðV �CÞ:

ð37Þ

Other material parameters such as reference remanent quantitiesand density are given below by

PR0 ¼ 0:40 C=m2; SR0a ¼ �1:0� 10�3; h0 ¼ 20 �C;

q ¼ 7800 kg=m3; SR0c ¼ 2:0� 10�3;

ð38Þ

where reference remanent polarization PR0 and reference remanentstrain SR0c are determined so that predicted values are close to ob-served values in literature. The value of SR0a was determined fromthe isochoric assumption of domain switching at the reference tem-perature, i.e., from SR0c þ 2S

R0a ¼ 0 at h = h0. Finally, we need the val-

ues of material parameters in the evolution equation of domainswitching (32). It has been widely recognized that the magnitudeof coercive field decreases with increase in temperature. Here theassumption of linear variation of material properties over tempera-ture given by (21) is also applied to the parameters in the evolutionlaw (32). The value of critical driving force f 90�c0 for 90� domainswitching at the reference temperature h0 is determined so thatthe calculated hysteresis loop fits with that of experiments givenby manufacturers. Then the value of a90�fc is determined so that the

http://www.ctscorp.comhttp://www.parameter.se/products/pgfiles/5423de2b-54b5-4427-b4d5-a9b5dcb885d0.pdfhttp://www.parameter.se/products/pgfiles/5423de2b-54b5-4427-b4d5-a9b5dcb885d0.pdfhttp://www.parameter.se/products/pgfiles/5423de2b-54b5-4427-b4d5-a9b5dcb885d0.pdf

Fig. 2. Effects of compressive stress on electric displacement (hysteresis loops) and strain (butterfly loops) responses of a ferroelectric polycrystal to cyclic electric fieldloading at three different temperatures, 20, 70, and 120 �C.


calculated hysteresis loops at room and elevated temperatures arequalitatively consistent with experimental observations. The criticaldriving force for 180� domain switching is set to be f 180�c0 ¼ 2f 90

�c0 at

the reference temperature h0, as done by Pathak and McMeeking(2008), Kamlah et al. (2005). In order to keep the same relation be-tween critical driving forces for 90� and 180� domain switching, it isassumed that a180�fc ¼ 2a90

�

fc in the temperature range of interest. Thevalue of creep exponent m is assumed to be the same for both typesof domain switchings; the value of am was chosen so that the calcu-

lated hysteresis loops are qualitatively consistent with experimen-tal observations. The value of _n0 was chosen to be 1 for both 90� and180� domain switchings:

f 90�

c0 ¼ 0:5� 106 J=m3; m0 ¼ 5:0;

a90�

fc ¼ �0:2� 104 J=ðm3 �CÞ; am ¼ �0:04=�C;

f 180�

c0 ¼ 1:0� 106 J=m3; _n0 ¼ 1:0=s;

a180�

fc ¼ �0:4� 104 J=ðm3 �CÞ:

ð39Þ


The material parameters in (39) are determined so that model pre-dictions fit well with measurements in the qualitative point of view.Their variations over temperature change are not clearly under-stood yet. First of all, compressive stress experiments at various ele-vated temperatures have to be made to understand the temperaturedependence of critical driving force for 90� domain switching. Thenelectric field-induced switching behavior at different temperaturesshould be analyzed carefully to characterize the temperature-dependence of critical driving forces, creep exponents, and so on.

Fig. 3. Effects of temperature on electric displacement (hysteresis loops) and strain (butthree different stresses, 0, �50, and �150 MPa.

Now we calculate the responses of the model developed in pre-vious sections and compare the simulations with available resultsfrom experiments on ferroelectric ceramics. First, the responses ofthe polycrystal material model under sinusoidal electric fieldcycling applied in the direction opposite to initial poling at variouscompressive stresses and temperatures are calculated. The ampli-tude and frequency of electric field cycling are Eamp = 2.15 MV/mand fE = 0.1 Hz, respectively. The magnitudes of constantcompressive stress are 0, �50, and �150 MPa; and surrounding

terfly loops) responses of a ferroelectric polycrystal to cyclic electric field loading at

a b

Fig. 4. Strain responses of (a) poled and (b) unpoled ferroelectric polycrystals to uniaxial compressive loading at 20, 70, and 120 �C.

Fig. 5. Electric displacement responses of a poled ferroelectric polycrystal touniaxial compressive loading at 20, 70, and 120 �C.


temperatures are 20, 70, and 120 �C. The compressive stress is ap-plied instantly at the beginning time of calculation and it remainsconstant until the calculation is finished. The initial state of thematerial before each computation is that each single crystal inthe polycrystal model is considered to have equal volume fractionsof six distinct types of domains. The response curves at maximumcompressive stress�150 MPa in Figs. 2 and 3 were obtained duringthe second electric field cycling due to the presence of partiallyunsymmetrical responses at the first electric field cycling.

The effects of compressive stress on the hysteresis and butterflyresponses of the model at three different temperatures 20,70, and120 �C are shown in top, middle, and bottom graphs of Fig. 2. Theeffects of compressive stress at room temperature 20 �C shown inthe top of Fig. 2 are familiar and qualitatively consistent withexperimental observations in Lynch (1996), Mauck and Lynch(2003), Zhou et al. (2005), where the height of hysteresis loop isdecreased with increase in the magnitude of compressive stress.With the presence of large compressive stress, the domains thatare polarized in the direction perpendicular to compressive stressbecome more stable and they have a tendency not to join domainswitching by electric field loading. Similar decrease in the height ofhysteresis loop with increasing compressive stress is observed atother temperatures too in Fig. 2. Looking at the right parts ofthe figure, it is observed that the magnitude of actuation strainin butterfly strain curves, i.e., the difference between maximumand minimum strains in butterfly strain curves, increases at�100 MPa and then it decreases significantly at �150 MPa. Thesmall increase and significant decrease in actuation strain withincrease in the magnitude of compressive stress was also observedexperimentally in Fig. 6 of Zhou et al. (2005) at room temperature.Similar changes of actuation strain with compressive stress are ob-served at higher temperature 70 �C. However, it is not so at 120 �C.The magnitude of actuation strain at �150 MPa is very small at70 �C but it is relatively large at 120 �C. It is because the compres-sive stress of �150 MPa is large enough to suppress any 90� do-main switching at 70 �C but it is not so at 120 �C. At 120 �C,electric field loading of 2 MV/m is strong enough to activate 90�domain switching even at the maximum compressive stress�150 MPa. The 90� domain switching at 120 �C and �150 MPa bythe large electric fields of about 2 MV/m can also be seen in the leftlower plot of Fig. 2.

The calculated responses shown in Fig. 2 are rearranged in Fig. 3so that it may show the effects of temperature on the hysteresisand butterfly responses of the model at three different magnitudesof compressive stress. The top graphs in Fig. 3 show the effects of

temperature at zero compressive stress. Both the height and widthof hysteresis loops decrease with increase in temperature at 0 MPa.In strain responses, the magnitude of actuation strain seems to de-crease with increase in temperature at zero compressive stress,though it is not significant. The calculated variations of polariza-tion hysteresis and strain butterfly responses over temperature at0 MPa are qualitatively consistent with experimental observationsof Chong et al. (2008), Kounga et al. (2008), Kim (2010), Yimnirunet al. (2007), and aixACCT (www.aixacct.com, see Senousky et al.(2009)). But the behavior of calculated strain butterfly responsesat larger compressive stresses �100 or �150 MPa is totally differ-ent from that at 0 MPa. At larger compressive stresses, the magni-tude of actuation strain increases with increase in temperature,which is opposite to the behavior at 0 MPa. More experimentalinvestigations are needed to verify this strain behavior at hightemperatures.

Next we calculate the responses of the polycrystal model touniaxial compressive stress at three temperatures 20, 70, and120 �C. The strain and electric displacement responses shown fromFigs. 4–6 are calculated when an applied compressive stress isdecreased down to �296 MPa and then increased to zero at therate of 3.7 MPa/s. The calculated responses in the figures can becompared qualitatively well with the experimental observationsof Webber et al. (2009).

http://www.aixacct.com

Fig. 6. Total, linear and remanent strain and electric displacement responses of a poled ferroelectric polycrystal to uniaxial compressive loading at 20, 70, and 120 �C.


The calculations in Fig. 4(a) and (b) show the strain responses ofpoled and unpoled polycrystals, respectively, to uniaxial compres-sive stress. At 20 �C, strain increases linearly at a relatively smallstress but it shows a nonlinear increase when the magnitude of ap-plied stress becomes larger than about 60 MPa. The nonlinear ef-fect disappears when no more domain is available for ferroelasticdomain switching, which occurs at about �180 MPa in Fig. 4(a).After that, the material behaves as a linear material and the strain

curve becomes a straight line, until the stress is reduced to zero. Inan unpoled material, the amount of domains available for ferro-elastic domain switching is smaller than that in a poled material.Thus the magnitude of switching-induced strain is smaller in apoled material than in an unpoled material. It is observed inFig. 4 that the qualitative features of strain responses at hightemperatures 70 and 120 �C are similar to that at 20 �C. However,the behavior of stress-induced domain switching is strongly


dependent on temperature. For example, nonlinear behavior due toferroelastic switching starts at a lower level of compressive stressat higher temperatures. The value of initial strain in Fig. 4(a) whenno stress is applied to the material looks the same at three temper-atures. It is because the chosen value of thermal expansion coeffi-cient is relatively small and the differences in initial strain atdifferent temperatures are not shown in the scale of the figure.In a smaller scale, it can be seen that the magnitude of initial strainincreases with increase in temperature. On the other hand, it iseasily observed in Fig. 4(a) that the magnitudes of maximum andremanent strain are decreased with increasing temperature.

The electric displacement responses of a poled polycrystal touniaxial compressive stress is shown in Fig. 5. The magnitude ofinitial electric displacement is decreased with increase in temper-ature due to negative pyroelectric coefficient in the crystallo-graphic c direction in (35). Earlier ferroelectric domain switchingat higher temperatures is also observed in Fig. 5. The calculatedstrain and electric displacement responses in Figs. 4 and 5 can becompared qualitatively well with the observed behavior in Figs. 3and 4 of Webber et al. (2009), where the longitudinal strain andpolarization of a PZT sample under uniaxial compressive stress atelevated temperatures were measured.

The calculated total strain shown in Fig. 4 can be decomposedinto linear strain and remanent strain at each temperature, as for-mulated in (27). The linear strain at zero electric field is obtainedby multiplying elastic compliance by applied stress; the remanentstrain is obtained by subtracting the linear strain from total strain.In the same way, the total electric displacement in Fig. 5 can be di-vided into linear and remanent electric displacement. Fig. 6(a) or(b) show total, linear, and remanent strains (or electric displace-ments) at three temperatures 20,70, and 120 �C. In the present cal-culation, elastic compliance of a tetragonal variant is taken to beisotropic and constant during temperature change, as shown in(35). Thus the linear strain curves in Fig. 6(a) become straight linesand the contribution of linear strain to total strain is constant inde-pendent of temperature change. The general features of linear andremanent strains in Fig. 6(a) are consistent with the experimentalobservations of Webber et al. (2009), Zhou and Kamlah (2010). Onthe other hand, piezoelectric coefficients are anisotropic anddependent on temperature in (36). As a result, the shapes of linearelectric displacement curves in Fig. 6(b) depend on temperatureand are no longer straight,which is qualitatively consistent withthe experimental observations of Zhou and Kamlah (2010). From(39) and (32), it is clear that the magnitude of critical driving forcefc decreases with increasing temperature, which is seen in Figs. 6(a)and (b). The rate of domain switching increases with increasingtemperature and mechanical depolarization is completed at astress level less than 50 MPa at the highest temperature 120 �C.It is depolarized so rapidly at 120 �C that linear electric displace-ment is nearly zero at almost all stress levels due to the absenceof piezoelectric coefficients.

5. Conclusion

A constitutive model for nonlinear behavior of ferroelectricpolycrystals in a relatively wide range of temperature is proposed.A ferroelectric polycrystal is composed of 210 single crystallites,each of which is orientated uniformly over all directions. Thebehavior of the polycrystal material model is determined byassuming the same electric, stress, and temperature fields for everyconstituent single crystallite and averaging the behavior of thecrystallites. A single crystallite is composed of six distinct typesof tetragonal lattices and each variant of a crystallite is character-ized by a Gibbs free energy function. In the present work, Gibbsfree energy function is derived directly from the thermodynamic

definitions of linear moduli instead of Helmholtz free energy func-tion and Legendre transformation. The Gibbs free energy potentialis a function of electric field intensity vector, stress tensor, andtemperature. Linear moduli in the free energy function are as-sumed to be either constant or linearly dependent on temperature.Application of the Clausius–Duhem inequality to a ferroelectricpolycrystal consisting of 210 single crystallites yields a dissipationinequality where the product of critical driving force and the rateof domain switching between two variants within a crystalliteshould be nonnegative. Domain switching between two differenttypes of variants in a crystallite is governed by an evolution equa-tion of viscoplasticity type. In the present model, the viscoplasticevolution law proposed by Huber et al. (1999) is modified so thatit may reflect experimentally observed temperature dependencyof coercive electric field and the shape of hysteresis loop. In themodified viscoplastic law, critical driving force and creep exponentare given as a linear function of temperature.

The proposed constitutive model is utilized to calculate the re-sponses of ferroelectric polycrystal to electric field and mechanicalstress loading. Electric displacement hysteresis loops and longitu-dinal strain butterfly curves are calculated at various magnitudesof compressive stress and temperatures. Effects of compressivestress on the shapes of hysteresis and butterfly loops are consistentwith previous observations, both loops becoming flatter at largercompressive stresses. However, as temperature is increased, do-main switching becomes more active and more variants participatein switching, resulting in thicker hysteresis and butterfly loops atlarge compressive stresses. The effects of temperature on the shapeof hysteresis and butterfly loops are consistent well with experi-mental observations available in literature, narrower loops due todecreasing coercive field and upward movement of butterfly loopsdue to thermal expansion effect with increasing temperature. Thethickness of hysteresis loop gets narrower with increasing com-pressive stress. But the magnitude of actuation strain increasedat �100 MPa and then a little decreased at �150 MPa. The calcu-lated responses of poled polycrystals to longitudinal compressivestress show that the magnitude of coercive stress decreases withincrease in temperature. The magnitude of remanent strain alsodecreases with increasing temperature. Mechanical depolarizationin unpoled polycrystals shows similar behavior, except smallermagnitude of remanent strain due to smaller amount of domainsavailable for ferroelastic switching. Total strain at each tempera-ture was decomposed into linear strain and remanent strain. Sinceelastic compliance is set to be isotropic and constant over temper-ature, the contribution of linear strain to total strain is the same forall temperatures. However, it is not so in case of electric displace-ment. Piezoelectric coefficient is temperature- and polarization-dependent and so the shapes of linear strain curves are not straightat all.

A constitutive model for ferroelectric polycrystals is proposed tosimulate the nonlinear thermo-electro-mechanical behavior of thematerials in a relatively wide range of temperature. Linear moduliand switching parameters are assumed to be linearly dependent ontemperature. It was observed that the responses of the model areconsistent qualitatively well with some experimental observations.However, it is suggested that the dependency of linear moduli andswitching parameters of 90� and 180� domain switchings ontemperature should be investigated furthermore. In addition, thecalculated behavior of the materials at high temperature has tobe compared with measured behavior that will be obtained fromfuture experiments.

Acknowledgements

This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded


by the Ministry of Education, Science and Technology (2010-0013040).

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A constitutive model for thermo-electro-mechanical behavior of ferroelectric polycrystals near room temperatureIntroductionThermodynamic frameworkThermo-electro-mechanical constitutive modelResults and discussionConclusionAcknowledgementsReferences

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