Accepted Manuscript

Rate-dependent Electro-mechanical Coupling Response of Ferroelectric Mate‐

rials: A Finite Element Formulation

Amir Sohrabi, Anastasia Muliana

PII: S0167-6636(13)00033-1

DOI: http://dx.doi.org/10.1016/j.mechmat.2013.02.005

Reference: MECMAT 2095

To appear in: Mechanics of Materials

Received Date: 7 September 2012

Revised Date: 18 January 2013

Please cite this article as: Sohrabi, A., Muliana, A., Rate-dependent Electro-mechanical Coupling Response of

Ferroelectric Materials: A Finite Element Formulation, Mechanics of Materials (2013), doi: http://dx.doi.org/

10.1016/j.mechmat.2013.02.005

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1

Rate-dependent Electro-mechanical Coupling Response of Ferroelectric Materials:

A Finite Element Formulation

Amir Sohrabi and Anastasia Muliana*

Mechanical Engineering Department, Texas A&M University

*Corresponding author: amuliana@tamu.edu

Abstract

This paper presents a three-dimensional (3D) constitutive model for predicting nonlinear

polarization and electro-mechanical strain responses of ferroelectric materials subject to various

histories of electric fields and mechanical stresses. The electro-mechanical coupling constants

are expressed as functions of a polarization state and it is assumed that in absence of the

polarization, the material does not exhibit electro-mechanical coupling response. The

polarization model due to an electric field input is additively decomposed into time-dependent

reversible and irreversible parts. The model also incorporates the effect of compressive stresses

on the polarization response. Thus, the constitutive model is capable of incorporating the effect

of loading rates, mechanical stresses, and electric fields on the overall hysteretic electro-

mechanical and polarization switching response of ferroelectric materials. The constitutive

model is implemented in a continuum 3D finite element and used to perform rate-dependent

electro-mechanical coupling analyses of smart structures. The experimental data on the

polarization switching and hysteretic butterfly strain responses of lead zirconate titanate (PZT)

reported by Fang and Li (1999) are used to validate the constitutive model. Parametric studies

are also conducted to examine the effect of loading rates and coupled electro-mechanical

boundary conditions on the overall performance of PZT. Finally, FE analyses are performed to

simulate shape changing in smart composite structures.

2

1. Introduction

Ferroelectric materials, such as lead zirconate titanate (PZT) and polyvinylidene fluoride

(PVDF), have been widely used in sensor, actuator, and energy conversion devices, where they

are subjected to various histories of electro-mechanical stimuli. Several experimental studies

have been conducted on understanding the response of ferroelectric materials under cyclic

electric fields with amplitudes higher than the coercive electric field limits. Under such loading

conditions, the ferroelectric materials exhibit polarization switching, e.g., Schmidt (1981),

Gookin et al. (1984) and Fang and Li (1999). It was also shown that compressive stresses that

are applied along the poling axis of the ferroelectric materials could induce depolarization of the

poled ferroelectric materials (Lynch (1996), Chen and Lynch (1998), Fang and Li (1999)). Fang

and Li (1999) experimentally studied changes in the polarization and butterfly strain loops of a

PZT specimen under a cyclic electric field input. After several cycles, the saturated polarization

response converges to a constant value, which is slightly smaller than the one measured in the

first cycle. An experimental study on a polarized PZT specimen under cyclic electric fields with

the maximum amplitude of 85% of the coercive electric field of the PZT, reported by Crawley

and Anderson (1990), shows nonlinear electro-mechanical response. They observed that the

effects of creep and loading rate on the piezoelectric constant were more significant at larger

strains and lower frequencies. The electrical and mechanical responses of ferroelectric materials

are time and frequency dependent, which were experimentally shown by Fett and Thun (1998),

Schaeufele and Hardtl (1996), Zhou and Kamlah (2005, 2006), Ben Atitallah et al. (2010),

among others. Zhou and Kamlah (2005, 2006) showed the creep response in a soft PZT under

static electric fields and compressive stresses, which were more pronounced at higher stresses

and at electric fields near the coercive electric field.

3

There have been constitutive models developed to predict nonlinear electro-mechanical

behaviors of ferroelectric materials, which can be classified as phenomenological (macroscopic)

models based on continuum mechanics approach and micromechanics based models that

incorporate the microstructural morphologies of the materials. In an analogy to rate-independent

plasticity theory, macroscopic constitutive models have been formulated for predicting

polarization switching response in ferroelectric materials due to electric field inputs. The strains

and electric displacements are additively decomposed into reversible and irreversible

components. Examples of these macroscopic models can be found in Bassiouny et al. (1988a and

b, 1989), Huang and Tiersten (1998a and b), Kamlah and Tsakmakis (1999), Linnemann et al.

(2009). Recently, Muliana (2011) presented a phenomenological model for time-dependent

polarization and electro-mechanical strain responses of ferroelectric materials subject to various

histories of electric fields. The model is capable of predicting the polarization switching response

of piezoelectric ceramics at various rates of electric field inputs. Massalas et al. (1994) and Chen

(2009) presented nonlinear electro-mechanical constitutive equations for materials with memory-

dependent (viz. viscoelastic materials). They also incorporate the dissipation of energy due to the

viscoelastic effect, which is converted into heat. The macroscopic response of materials depends

strongly upon their microstructural response, which occurs at various length scales.

Microscopically motivated constitutive models that take into account polarization response of

each crystal in predicting the overall nonlinear electro-mechanical response of ferroelectric

materials can improve our understanding on the nonlinear behavior of ferroelectric materials.

Examples of the micromechanics based constitutive models for polarization switching in

ferroelectric materials can be found in Chen and Lynch (1998), Fan et al. (1999), Li and Weng

(1999, 2001), Smith et al. (2003, 2006), Su and Landis (2007).

4

Finite element (FE) methods have been used for analyzing the electro-mechanical

response, including the hysteretic polarization switching response, of structures consisting of

conductive and ferroelectric materials mainly for time (rate)-independent behavior, e.g. Kamlah

and Bohle (2001), Landis (2002), Zeng et al. (2003), Li and Fang (2004), Zhang et al. (2005),

Klinkel (2006), Wang and Kamlah (2009), Linnemann et al. (2009), Klinkel et al. (2011), and

Muliana and Lin (2011). In the above FE formulations, macroscopic constitutive models are used

for the electro-mechanical coupling response of piezoelectric and ferroelectric materials. Zeng et

al. (2003) presented incremental and iterative solutions for problems involving polarization

switching due to high electric field and heat generation from the dissipation of energy during the

domain reversal process. FE methods that also include the time (rate) - dependent effects are

currently limited. Kim and Jiang (2002) presented FE algorithm for simulating macroscopic

polarization and strain responses in ferroelectric materials undergoing domain switching. The

macroscopic electro-mechanical constitutive models include the effect of different polar axes in

the crystallites, whose contributions are quantified by mass fractions. The macroscopic

polarization, strains and electro-mechanical properties are obtained using the weighted average

of their microstructural configurations through the mass fractions. They also defined the

functions for the rate of change of the mass fractions, which allow for incorporating rate-

dependent loadings.

Experimental studies show that the electro-mechanical response of ferroelectric materials

is time- (and rate-) dependent when subjected to electric fields and mechanical loadings. The

electro-mechanical response of ferroelectric materials also depends strongly upon the applied

electric fields and mechanical stresses. This study presents a three-dimensional (3D) rate-

dependent electro-mechanical coupling constitutive model for ferroelectric materials undergoing

5

various histories of mechanical stress and electric field. The constitutive model is derived for

materials undergoing small deformation gradients which are suitable for ferroelectric ceramics.

The constitutive model is capable of predicting the overall electro-mechanical and polarization

switching behaviors of the ferroelectric materials. This study concerns with the polarization

switching response due to application of electric fields and examines the effect of mechanical

stresses during the polarization switching. It is noted high stresses can also induce polarization

switching in ferroelectric materials; however, stress induced polarization switching is not being

considered in this manuscript. The polarization due to an electric field input is additively

decomposed into the time-dependent reversible and irreversible parts, in which the irreversible

part is due to the polarization switching process. The electro-mechanical coupling material

constants are taken as nonlinear functions of a polarization state and in absence of the

polarization the electro-mechanical coupling constants would vanish. The model also assumes

that the coercive electric field of the ferroelectric material depends on the mechanical stress. This

constitutive model is implemented in 3D continuum element and used to perform structural

analyses. Two scales of integration algorithms based on recursive and iterative schemes are

formulated at the constitutive material and finite element levels. The manuscript is organized as

follows. Section 2 presents the time-dependent electro-mechanical constitutive model followed

by its FE implementation in the three-dimensional continuum elements in Section 3. Section 4

presents numerical examples of the time-dependent nonlinear electro-mechanical response of

ferroelectric structural components. Section 5 is dedicated to concluding remarks.

2. Constitutive Model

2.1 Time-dependent polarization response

6

This section presents the time-dependent polarization model of ferroelectric ceramics that belong

to the Perovskit polycrystalline structures. Figure 1 illustrates a schematic representation of a

polycrystalline structure including the grain boundaries and crystallites. At temperature below

the Curie temperature each crystal forms a tetragonal shape due to a spontaneous separation of

the positive and negative charges, known as electric dipole1. This separation is measured by a

dipole moment which is a vector quantity and a dipole moment per unit volume is known as

polarization. In Fig. 1, an arrow inside each crystal indicates the direction of the spontaneous

polarization. When the directions of the polarization in the crystallites are distributed randomly,

the ferroelectric materials are macroscopically non-polarized (Fig. 1a) which is denoted by the

polarization P3=02. By applying a sufficiently high electric field the direction of the spontaneous

polarization in the crystallites can be reoriented towards the direction of the external electric field

so that the ferroelectric materials are macroscopically polarized (Fig. 1b). It is observed that the

macroscopic polarization response of ferroelectric based materials depend also on the rates

(frequencies) of the external electric field applied (Ben Attitalah et al. 2010). The polarized

ferroelectric materials can be depolarized by applying an electric field in the opposite direction to

the current poling direction or by prescribing a compressive stress along the poling axis.

The time-dependent hysteretic polarization model (Muliana 2011) is adopted to describe

the macroscopic polarization response of ferroelectric materials due to various histories of

external electric field inputs and modified to include the effect of the mechanical stress on the

polarization response of the ferroelectric materials. Consider an electric field3 input in the x3

1 Please see Lines and Glass (2009) and Ballas (2007) for a detailed explanation. 2 The macroscopic polarization is considered in the x3 direction with regards to the Cartesian coordinate system. 3 The electric field is defined by ,i iE where is the electric potential.

7

direction 3( )E s , 0s and 3( ) 0, 0E s s , where s is the time history. The corresponding

polarization response at the current time t is:

3 3 3 3 3[ ( ), ] [ ( ), ] [ ( ), ]tP P E t s t R E t s t Q E t s t (2.1)

where 3( ),R E t s t is the time-dependent reversible polarization response at current time 0t

with 0, 0R t and 3( ),Q E t s t is the residual (irreversible) polarization. The reversible

polarization response is expressed as:

0 33 3 3

30

[ ( ), ] [ , ] [ , ] 0

t st sR dE

R R E t s t R E t E t s ds tE ds

(2.2)

0 0 0

3 0 3 1 3

1

[ , ] ( ) ( ) 1. expt

R E t R E R E

(2.3)

One may consider 0

3 ,R E t as the polarization at current time t due to a constant electric field

applied at s=0. The superscript s and t denote the representative of the previous time history and

current time, respectively. Both 0 3

sR E and 1 3

sR E are functions of 3

sE . The characteristic time4

1 measures the speed of the polarization changes with time. In a linear case 0 3

sR E and 1 3

sR E

are considered as follows:

0 3 0 3

1 3 1 3

( )

( )

s s

s s

R E E

R E E

(2.4)

where 0 is the dielectric constant of a macroscopically non-polarized material, corresponding to

the second order permeability tensor in a multi-axial case; 1 is the time-dependent part of the

dielectric constant and when 1 0 , a time-independent polarization behavior is considered.

4 The rate of polarization depends on the electric field applied, as shown by Zhou and Kamlah (2006). This effect

can be incorporated by taking the characteristic time to depend on the electric field.

8

The state of polarization is defined through the following polarization function:

2 2

3 3,t t

c cf P P P P (2.5)

where cP is the current polarization state, analogous to yield stress in overstress plasticity theory.

It is assumed that the irreversible polarization is formed when 3( , ) 0t

cf P P and 3 3 0t tE P . The

irreversible polarization is defined as:

3

3 3

30

[ ]

tE st t sdQ

Q Q E dEdE

(2.6)

33

3

3

3

, 0

exp 1 , 0

0 0

nt

t

c

c

ttt

c

c

EE E f

E

EdQE E f

dE E

f

(2.7)

where cE is the coercive electric field and , , ,n are the material parameters that are

calibrated from experiments. In a non-polarized sample, the current polarization state 0cP , and

once the ferroelectric sample is completely polarized the current polarization state is equal to the

saturated polarization ( sc PP or c sP P ). During loading, indicated by 3 3 0t tE P , the

polarization state will be updated by taking 3

t

cP P if the polarization function satisfies

3( , ) 0t

cf P P and a new polarization state is determined. When the materials undergo unloading

or neutral loading, 3 3 0t tE P , or loading that results with 3( , ) 0t

cf P P , there will no further

update on the polarization state. The above equations are solve numerically (discussed Section

3), in which the rate of polarization is approximated as: 33

tt P

Pt

.

9

2.2 Three-dimensional constitutive model

Ferroelectric materials exhibit macroscopic electro-mechanical coupling response when they are

macroscopically polarized. This is shown by an elongation in the material along the electric field

line and a contraction in the transverse directions when the electric field is applied in the poling

direction. When the electric field is applied opposite to the poling direction, the material

experiences shortening along the electric field line and expansion in the transverse directions and

when the electric field is applied perpendicular to the poling directions, the transverse shear

deformations are shown. The macroscopic strains due to the polarization are measured through

the piezoelectric constant 3( )tPg whose magnitude depends on the polarization state. The

polarized ferroelectric material could experience depolarization 3 0tP when a sufficient electric

field is applied in the opposite direction to its poling axis. When the depolarization occurs, the

materials loss their electro-mechanical coupling effect, which is represented by (0) 0g .

Polarizing the ferroelectric material by applying an electric field, while at the same time the

ferroelectric material is under a compressive stress along the electric field line, results in

reductions of the saturated and remanent polarizations and the coercive electric field.

A nonlinear electro-mechanical coupling constitutive model for ferroelectric ceramics

undergoing small deformations that incorporates changes in the polarization due to an electric

field while undergoing mechanical stresses is5:

4

2

t t t t t t t

ij ijkl kl nij nm mkl kl kij k

t t t t

i im mkl kl i

S g g g P

D g P

(2.8)

5 Under a sufficiently high compressive stress the mechanical strains of the ferroelectric ceramics show nonlinear

and inelastic response (Fang and Li, 1999). This can be included by considering an inelastic constitutive model for

the first term of the strain component in Eq. (2.8), see Muliana (2010).

10

where ijklS is the scalar component of the elastic compliance tensor measured at fixed electric

field, , ,t t t

ij i iD P are the scalar components of the mechanical stress, electric displacement and

polarization, respectively, ij is the scalar component of the permittivity constant of a polarized

specimen measured at fixed stress and constant (remanent) polarization rP , and the small strain

is defined as , ,

1

2ij i j j iu u , where iu is the scalar component of the displacement. The scalar

component of the piezoelectric constant t

ijkg depends on the current polarization state 3

tP :

3 1/

33( )

ttP Ct t r

ijk ijk ijk

r

Pg g P e g

P

(2.9)

where rP is the remnant polarization, 1C is the material parameter that needs to be calibrated

from experiment (see Muliana 2011), and r

ijkg is the scalar component of the piezoelectric

constant measured at constant polarization rP . Thus, the constitutive model in Eq. (2.8) is a

nonlinear function of electric field and depends on time. The third component of the polarization

3

tP in Eq. (2.8) is given in Section 2.1, while the other two components are

1 11 1

t tP E and 2 22 2

t tP E (2.10)

Experimental studies show that the coercive electric field of ferroelectric materials depends

on the compressive stress applied to the material along its poling direction, while little is known

about the effect of tensile stress on the polarization response of ferroelectric ceramics. This is

due to the fact that ceramics is brittle and has a relatively low ultimate strength under tension. It

then is necessary to have the coercive electric field varies with the compressive stresses and we

further assume that the coercive electric field remains unaltered with the tensile stress:

11

33 33

33

, , 0.0

, 0.0

o t t

c c

c o t

c

E EE

E

(2.11)

where o

cE is the coercive electric field in absence of the mechanical stresses. The existence of

compressive stresses also influences the polarization response of ferroelectric materials. When a

compressive stress higher than the coercive stress c is applied to the polarized ferroelectric

ceramics, the materials undergo the polarization switching (Fang and Li 1999). Here, we

consider the mechanical stress is first prescribed to the unpolarized samples and this stress is

kept constant while a cyclic electric field is applied. Our attempt is to incorporate the effect of

compressive stress on the overall polarization switching response. It is assumed that the

compressive stress that is higher than the coercive stress limit affects the polarization state 3

tP

and the piezoelectric constants:

3 1 2 33/ /3

3

2 33

( )

0 when

t tc

tP C Ct t r

ijk ijk ijk

r

t

c

Pg g P e e g

P

C

(2.12)

where the material parameters 2, c C have positive values and they need to be calibrated from

the experimental tests.

3. Finite Element Formulation

A three-dimensional (3D) continuum finite element for nonlinear time-dependent electro-

mechanical response is formulated. The following field variables: displacements in the three

directions of the Cartesian coordinate system and electric potential are sampled at each node

within a finite element. Here 1 1 1

1 2 3 1 2 3, , ,..., , ,n Nd Nd Ndu u u u u uUT

and 1 2, ,...,n Nd ΦT

are

the nodal displacement and electric potential vectors, respectively, in a single element with a

12

number of nodes Nd. The mapping of the field variables is done through the use of shape

functions 1 2, ,..., NdN N N :

1

1

1,2,3Nd

i i

k k

i

Ndi i

i

u N u k

N

(3.1)

The corresponding strains and electric fields, which are sampled at the material integration points

within the finite element, are obtained as:

, ,

1

2

u n

ij i j j i ijm mu u B U or u nε B U (3.2)

n

i im m

i

E Bx

or

nE B Φ (3.3)

where uB and

B are the spatial derivative of the shape functions related to the macroscopic

strain and electric field, respectively. The stress and electric displacement counterparts are

determined using the constitutive relation discussed in Section 2. The overall governing

equations for the electro-mechanical deformation are formed at the structural level by imposing

the energy balance equations. In this study, the nonlinear time-dependent electro-mechanical

constitutive model is expressed in terms of the stress and electric field components as the

independent field variables, while the displacement based finite element formulation leads to

strain and electric field components as the independent variables. Thus, the constitutive model in

Section 2 cannot be directly implemented in the finite element. The nonlinear time-dependent

response is solved incrementally by linearizing the response and iteratively correcting the

residual (error) from the linearized solutions. It is necessary to define the consistent tangent

stiffness, piezoelectric and dielectric constants.

13

3.1 Time-integration algorithm at the material level

The electro-mechanical constitutive model, Eq. (2.8), depends on the polarization state 3

tP , which

is a function of the electric field input 3

sE . A time-integration algorithm is formulated based on a

recursive method in order to approximate the current polarization state. At each time t, the

polarization state is approximated as:

3

3 3 3 3

3

; =

;

t t t t t t t

tt t t t t t

P R Q Q Q Q

dQQ E E E E

dE

(3.4)

where the superscript t t denotes the previous time, tQ is the incremental irreversible

polarizations, and t tQ is the history variable defining the irreversible polarization at the

previous converged time. The reversible polarization in Eq. (2.2) is approximated as:

0 3 1 3 30 3 1 3

1 3 3 10

0

0 3 1 3 1 3

1

( ) ( )( ) ( ) 1 exp 1. exp

( ) ( ) ( )exp

t s s st t t

t t t

t R E R E t s dER R E R E ds

E E ds

tR E R E R E q

(3.5)

where the history variable related to the reversible polarization is:

1 3 3

3 10

1 3 3 1 3 3

3 1 3 10

( )exp

( ) ( ) exp exp

t s st

t t ts s s s

t t

R E t s dEq ds

E ds

R E t s dE R E t s dEds ds

E ds E ds

(3.6)

1 3 3 1 3 3

1 3 1 3

( ) ( )exp exp

2

t t t t t tt t tt R E E t R E E t

q qE t E t

(3.7)

At an initial time, 0 0.0tq q and 0 0

0( )R R E .

14

Once the polarization state is determined the electro-mechanical response of the ferroelectric

materials at current time can be determined. Within an incremental time t, the incremental

nonlinear constitutive relation in Eq. (2.8) can be expressed in a linearized form:

't t tT

t

t t t

ε σ σS dA

D E Ed κ (3.8)

The consistent tangent compliance6, piezoelectric, and dielectric constants are defined as:

' 3

3

4

2

t t t t tij ij kijt t t t m

ijkl ijkl nij nm mkl kij m mijt t t

kl k m k

t tti i

ikl im mkl ij

kl j

g P PS S g g d P g

E P E E

D Pd g

E

(3.9)

where the partial derivative of the nonlinear piezoelectric constant and polarization in Eq. (3.9)

are expressed as:

3

13

3 1

31 2 311 22 0 1 3 3

1 2 3 3 3

1

; ; 0.5

tPt tkij C r

kijt

r r

t

tt t t tt t t

t t t t t

g Pe g

P P C P

dQ

dEP P P dQE E

E E E E dE

(3.10)

As mentioned above, the constitutive model is implemented in a displacement based finite

element framework, in which the strain and electric field variables are taken as the independent

variables obtained from Eqs. (3.2) and (3.3) and the corresponding stresses and electric

displacements need to be determined. The incremental strain and electric field at current time are

6 If a nonlinear stress-dependent constitutive model is considered for the mechanical strain in Eq. (2.8), the

consistent tangent compliance is written as 4ij t t

ijkl nij nm mkl

kl

fS g g

, where f is the nonlinear mechanical

strain. In this manuscript we ignore the inelastic mechanical response and consider the linearized elastic mechanical

response for the studied ferroelectric ceramics. This is done in order to highlight the nonlinear and irreversible

behaviors due to polarization switching.

15

obtained from , ,t u n t u n t t ε B U B U and , ,t t t t E B U B U , respectively. For this

purpose, the linearized constitutive relation in Eq. (3.8) is rewritten as:

'

'

t t tT

t

t t t

σ ε εC eB

D E Ee κ (3.11)

where the consistent tangent stiffness, piezoelectric constant, and dielectric constants are:

1 1

1 1

' '

' '

T

T

TC S e S d

e dS κ κ dS d (3.12)

Finally the stresses and electric displacements at current time are:

t t t t

t t t t

σ σ σ

D D D (3.13)

The polarization state 3

tP is a function of the coercive electric field that depends on the

current compressive stress 33

t . In the displacement based FE, the current value of stresses need

to be determined from the strain and electric field inputs. In this study, during the incremental

solution at the material level, the coercive electric field at current time is obtained as:

33 33

33

, , 0.0

, 0.0

o t t t t

c c

c o t t

c

E EE

E

(3.14)

Thus, the calculated consistent tangent stiffness, piezoelectric constant, and dielectric constants

in Eq. (3.12) depend on the current electric field 3

tE and stress from the previous time increment

33

t t . Instead of performing iteration at the material level, the correction due to a linearized

stress is performed through an iteration scheme at the structural level.

3.2 Solution at the structural level

16

The governing equations for the electro-mechanical deformation under a quasi-static loading and

small-deformation gradients are formed at the structural level by imposing the energy balance

equations, which in absences of the body forces and body charges are:

t t t t

ij ij i i

V A

t t t t

i i s

V A

d dV t du dA

D dE dV q d dA

or

T T

V A

T

s

V A

d dV d dA

d dV d q dA

ε σ u t

E D (3.15)

where t and qs are the surface traction and surface charge, respectively. Using the strain and

electric field defined in Eqs. (3.2) and (3.3), respectively, the linearized constitutive model in Eq.

(3.11), and the principle of virtual work, the energy balance equations for one element are:

'

'

nT uT u n T n nT T

V A

nT T u n n nT T

s

V A

d dV d dA

d dV d q dA

U B CB U e B Φ U N t

Φ B eB U κ B Φ Φ N (3.16)

Equation (3.16) can be rewritten as:

'

'

nT uT u n uT T n nT T uu n u n M

V V A

nT T u n T n nT T u n n E

s

V V A

d dV dV d dA

d dV dV d q dA

U B CB U B e B Φ U N t K U K Φ F

Φ B eB U B κ B Φ Φ N K U K Φ F (3.17)

In order to obtain solutions for the displacements and electric potential at the element level the

Gaussian quadrature method is used for the spatial integration. At each time increment the

linearized relations in Eqs. (3.11)-(3.13) are used as a starting point for obtaining trial solutions

to the nodal displacements and electric potential. The Newton-Raphson iterative method is then

used to correct for the errors from the linearization. After assembly over all elements the overall

equilibrium equation can be obtained with the residual vector at time t:

, , , , ,,

, , , , , ,

M t uu t n t u t n tu t

t

t E t u t n t t n t

F K U K ΦRR

R F K U K Φ (3.18)

17

The above equation is solved when the boundary conditions are prescribed to the structures:

u

on S

on S

on S

on S

t

s qq

t σn

Dn

u u (3.19)

where n is the unit outward normal vector on the boundaries tS and qS . It is also necessary for

the boundaries to satisfy the following conditions:

u uS S S and S S 0

S S S and S S 0

t t

q q

(3.20)

It is noted that at the element level, the nodal displacements and electric potentials are

taken as the independent field variables , ,t n t n tX U Φ and in order to minimize the

residual vector at each time due to the trial linearized solution, the independent field variables

need to be corrected. Let k be an iterative counter, the corrected field variables at time t is:

1

,, 1 , ,

t kt k t k t k

RX X

XR (3.21)

and

, , , ,,

, , , ,

,

uu t k u t kt k

u t k t k

t k

K KR

X K KK (3.22)

The numerical algorithm at the structural and material levels within each time increment is

summarized as follows:

1. Input variables: , , , ,, , , , , , ,n t t n t t M t t E t t t t t t t t

cQ q P U Φ F F K

2. Determine trial nodal variables at time t: , , , ,, ; 0n t k n t k k U Φ

3. Iterate for k=0,1,2,… (k=iteration counter)

18

3.1. Calculate , , , , , , , , , , , , ,

3, , , , , , , , , ,t k t k t k t k t k t k t k t k M t k E t k t kP E g B σ D σ D F F K ;

3.2. Define residual (Eq. 3.17) ,t k

R and check for ,t k TolR ; yes then go to 4 else

3.3. Correct the nodal displacement and electric potential (Eq. 3.19) , , 1 , , 1,n t k n t k

U Φ and go

to 3.1

4. Output variables: , , , ,, , , ,n t n t M t E t t

U Φ F F K and update history variables , ,t tcQ q P .

4. Numerical Implementation

This section presents analyses of the electro-mechanical response of ferroelectric materials and

structural components undergoing coupled mechanical loading and electric field. Experimental

data on the polarization switching response of PZT 51, reported by Fang and Li (1999), are used

to validate the constitutive model. Parametric studies on understanding the effects of different

boundary conditions and loading rates on the electro-mechanical response of the ferroelectric

materials are presented. Finally the FE method is used to perform time-dependent analyses of

smart structural components undergoing various histories of mechanical loading and electric

fields.

4.1 Verification of the constitutive model

The electro-mechanical constitutive model in Eq. (2.8) is validated using the experimental data

of PZT51 reported by Fang and Li (1999). Figure 2a shows the polarization hysteretic response

(D3-E3) of PZT-51 subject to a cyclic electric field at zero stress. The amplitude of the electric

field is 1.2 MV/m with a frequency of 1 Hz. The test started from an unpolarized condition and

with increasing the electric field the polarization takes place. The loop in the first cycle is higher

19

by about 0.03 C/m2. After several cycles, the saturated polarization converges to a constant value,

which shows a slightly smaller value than the one in the first cycle. The material parameters for

the time-dependent polarization in Eqs. (2.3), (2.4), and (2.7) are calibrated from this hysteretic

polarization response. The time-dependent polarization model is shown to be capable in

capturing the hysteretic polarization response. The material parameters of the PZT-51 used in the

simulation are reported in Table 1. The corresponding butterfly hysteretic responses during the

first and saturated cycles at zero stresses are shown in Fig. 2b. It is seen from Fig. 2b that at the

coercive electric fields the strains from the experimental tests are nonzero, which is about 500

higher. This might be due to the accumulated strain from the first cycle loading. In this study,

the parameter C1 in Eq. (2.9) is calibrated from the saturated butterfly curve in Fig. 3a after

shifting the strain response obtained from the simulation 500 higher. The piezoelectric

constants measured at the remanent polarization are obtained from Fang and Li (1999) and the

permittivity constants at the remanent polarization are taken from Muliana (2010). It is noted that

the piezoelectric constants at the remanent polarization used in Eq. (2.8) are determined from

1r r rg κ d . Tables 2 and 3 report the electro-mechanical coupling parameters and elastic

constants, respectively, for PZT-51. The corresponding transverse butterfly strain response is

shown in Fig. 3b. The nonlinear electro-mechanical coupling model is capable of simulating the

hysteretic polarization switching electro-mechanical response.

Next, the hysteretic responses due to a cyclic electric field at various constant compressive

stresses are simulated. Figures 4 and 5 illustrate the polarization and butterfly strain responses

under a cyclic electric field with amplitude of 1.2 MV/m and frequency of 1 Hz and various

compressive stresses: 5-30 MPa. From the experimental evidences the coercive electric fields

vary with the compressive stresses, which can be described by the following function:

20

330.0041o t

c cE E (4.1)

The material parameters reported in Tables 1 and 2 are used in the simulation. Good model

predictions are shown for the response up to 20 MPa compressive stresses and at the

compressive stress 30 MPa, the model over-predicts the polarization and strain responses. The

effect of compressive stresses in reducing the overall axial strain and electric displacement is due

to the electro-mechanical coupling 33 33 334 t t t

n nm mg g and 3 33 332 t t

m mg , respectively, with indices

m and n vary from 1 to 3. Fang and Li (1999) discussed that when the ferroelectric materials are

subjected to compressive stresses higher than the coercive stress c of the materials, polarization

switching occurs, which can alter the material properties. Figures 4d and 5d indicate that under a

compressive stress of 30 MPa the material properties in PZT-51 change significantly which is

shown by a low strain response. In order to incorporate the effect of high compressive stresses on

the polarization switching due to cyclic electric field, the piezoelectric constants in Eq. (2.12) is

used. This study used the hysteretic responses under a compressive stress 40 MPa (Figs. 6b and

7b) to calibrate the material parameters due to high compressive stresses. The coercive stress is

taken as 25 MPa. In this study an attempt is made to recalibrate the parameters corresponding to

the irreversible polarization. Table 4 presents the calibrated material parameters above the

coercive stress limit. The hysteretic responses under compressive stresses 30-80 MPa are shown

in Figs. 6 and 7. Relatively good predictions of the polarization and butterfly strains are observed.

4.2 Parametric studies

The electro-mechanical coupling response of a PZT-51 plate under various prescribed electro-

mechanical boundary conditions and loading rates is examined. The PZT plate has a dimension

of 10x10x1 mm3 and an electric field is applied through the thickness of the plate, i.e., along the

21

x3 direction. The FE method with 3D continuum elements (Section 3) is used to solve the

boundary value problem. The first case considers the plate under the following histories:

33( ) MPat rt and 3( ) 1.2sin(2 ) MV/mE t t . The following boundary conditions are

prescribed to the PZT plate:

1 2 3 2 3

2 1 3 1 3

3 1 2 1 2 1 2

1 2 1 2

3 1 2

(0, , , ) 0.0 0 10 and 0 1

( ,0, , ) 0.0 0 10 and 0 1

( , ,0, ) ( , ,0, ) 0.0 0 10 and 0 10

( , ,1, ) 1200sin(2 ) 0 10 and 0 10

( , ,

u x x t x x

u x x t x x

u x x t x x t x x

x x t t V x x

t x x

331, ) ( )t t

(4.2)

Figures 8 and 9 show the stress and electric field input histories and the corresponding electric

displacement and axial strain responses under two different stress rates: 20/6 and 80/6 MPa/s,

respectively. The duration of loading is 6 seconds. In the first loading history, the maximum

compressive stress is 20 MPa, which is below the coercive stress limit of PZT-51. It is seen in

Fig. 8 that the hysteretic polarization and butterfly strain responses decrease with increasing the

compressive stress, which is expected. The butterfly strain response is also shifted down

progressively, which is due to the mechanical strain component 3333 33

tS . The second loading rate

leads to high compressive stress (Fig. 9), in which the polarization switching due to high

compressive stresses occurs during the second cycle of loading. This can be seen by a sudden

drop in the butterfly strain when the magnitude of the compressive stress exceeds 25 MPa.

Next, the effect of frequency on the hysteretic electro-mechanical response is examined. The

PZT plate is subjected to 3( ) 1.2sin(2 ) MV/mE t ft and different frequencies f: 0.5, 1, and 10

Hz are considered. The PZT plate is kept under a traction free boundary condition. The first four

of the boundary conditions in Eq. (4.2) are prescribed. Figure 10 depicts the hysteretic responses

at various frequencies. The lowest frequency, f=0.5 Hz, results in the highest polarization and

22

strain responses. This is due to time-dependent effect of the polarization, in which under a

constant electric field the polarization increase with time (creep-like effect). Thus, a relatively

slow loading with regards to the characteristics time 1 1sec gives sufficient time for the

polarization to experience „creep-like‟ behavior. It is also seen that under both frequencies 0.5

and 1 Hz, the responses reach a saturated condition after a few cycle since the total time needed

to complete for example 3 cycles of loading is 6 and 3 seconds, respectively. This total time is

relatively large compared to the characteristic time of the PZT-51. Under the frequency 10 Hz,

the response shows continuous changes with time even after 10 cycles since the duration of

loading is within the characteristic time of the material during which the changes in the material

properties with time are occurring.

4.3 Analyses of Smart Structures

The PZT-51 is now used to induce deformations in a fiber reinforced laminated composite plate.

Consider a composite plate (host structure) of 100x50x1 mm3, shown in Fig. 11, with four PZT

patches. Each PZT patch has a dimension of 10x10x1 mm3. The properties of the PZT are given

in Tables 1-4, while the composite plate is made of a glass fiber composite. The composite plate

is assumed linear elastic with the following elastic material properties: E=36000 MPa, =0.25 in

the longitudinal fiber axis. The plate is clamped along one of its 50 mm side surface and the PZT

patches are bonded perfectly to the host structure. The potential on the surfaces of the PZT

patches that are in contact to the host structure is grounded to zero. The PZT patches are

uniformly subjected to the potential gradient through their thickness 3( ) 1.2sin(2 ) MV/mE t ft

so that they experience expansion in their planar direction, thus inducing bending to the

composite plate. Figure 12 depicts the lateral displacement measured at the mid section of the

23

free end due to the input electric field applied uniformly to the four PZT patches, which show

vibration of the composite plate. It is seen that the highest deformation occurs at the first quarter

cycle, when the electric field reaches 1.2 MV/m (point A). Upon removal of the electric field, the

remanent deformation is shown (point B) and when the electric field reaches the coercive limit of

the PZT patches no deformation is shown in the plate (point C) due to depolarization of the PZT

patches. Continuing applying the electric field till it reaches the lowest peak -1.2 MV/m (point D)

trigger bending in the composite plate. It is also seen that due to the time-dependent effect, the

highest displacement is at the first quarter cycle and after several cycles, saturation in the

butterfly displacement loop is achieved. The corresponding deformed shapes of the composite

plate at several instant of times during the first cycle are illustrated in Fig. 13. It is also possible

to create different deformed shapes by considering various histories of electric field inputs. For

example, to induce twisting of the plate different electric field inputs can be applied to the four

patches. The tip-deflection, illustrated in Fig. 14, is generated by prescribing the following

electric field input 3( ) 1.2sin(2 ) MV/mE t ft to the PZT patches 3 and 4, and

3( ) 1.2sin(2 / 2) MV/mE t ft to the patches 1 and 2.

5. Conclusions

A 3D rate-dependent electro-mechanical coupling constitutive model has been formulated for

ferroelectric materials undergoing various histories of mechanical stress and electric field. The

constitutive model is suitable for predicting nonlinear response of ferroelectric ceramics

undergoing small deformation gradients, including hysteretic polarization and butterfly strain

responses. The electro-mechanical coupling constants are assumed to depend on the polarization

state and in absence of the polarization the materials do not exhibit electro-mechanical coupling

24

effect. The polarization due to an electric field input is additively decomposed into the time-

dependent reversible and irreversible parts. This constitutive model is implemented in 3D

continuum element and used to perform structural analyses. Integration algorithms based on

recursive and iterative schemes have been formulated at the constitutive material and FE levels.

Experimental data on the polarization switching response of a PZT material reported by Fang

and Li (1999) have been used to validate the present constitutive model. It is shown that the

constitutive model is capable of predicting the hysteretic polarization and butterfly strain

response under a sinusoidal electric field input and several constant compressive stresses. We

have also conducted parametric studies to examine the effect of various prescribed electro-

mechanical boundary conditions and loading rates on the overall electro-mechanical response of

PZTs. Finally, FE analyses have been performed on controlling the deformation in a smart

composite plate, comprising of a laminated composite for the host structure and PZT patches.

Different histories electric field inputs can be applied to the PZT patches in order to induce

desired deformation such as bending and twisting. Due to the rate-dependent response of PZT

materials, it is necessary to prescribe not only an appropriate magnitude of electric field but also

the rate of electric field input in order to control the deformation of the smart structure.

Acknowledgement

This research is sponsored by the Air Force Office of Scientific Research (AFOSR) under grant

FA 9550-10-1-0002 and the National Science Foundation (NSF) under grant CMMI-1030836.

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27

Tables

Table 1 Material parameters for the time-dependent polarization of PZT-51

o

cE 0 1 1 n

(MV/m) (x10-9

F/m) (sec) (x10-6

F/m) (x10-6

F/m)

0.67* 70 225 1.0 0.35 3.0 1.6 4 * from Fang and Li (1999)

Table 2 Electro-mechanical coupling parameters for PZT-51

333

rd 311

rd 11

r 33

r Pr C1

(x10-12

m/V) (x10-9

F/m) (C/m2)

1520* -570* 38 42 0.194* 0.19 * from Fang and Li (1999)

Table 3 Elastic constants for PZT-51 (Muliana 2010)

11 22E E 33E 12G 13 23G G

12 13 =23

(GPa)

34.48 33 13.19 12.37 0.307 0.334

Table 4 Material parameters above the coercive stress limit

c C n

(MPa) (x10-6

F/m) (x10-6

F/m)

25 0.3 0.4 3.0 1.1 4

28

Figures

Figure 1 A schematic representation of polycrystalline structures of ferroelectric ceramics

Figure 2 Hysteresis polarization and butterfly strain responses for PZT51 at zero stress

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1.5 -1 -0.5 0 0.5 1 1.5

D3 (C/m2)

E3 (MV/m)

a) Polarization response

First cycle

Saturated

Model

0

1000

2000

3000

4000

-1.5 -1 -0.5 0 0.5 1 1.5

33 )

E3 (MV/m)

b) Strain response

First cycle

Saturated

Model

a) Non-polarized sample

b) Polarized sample

3 0P 3 0P

29

Figure 3 Saturated strain responses for PZT51 at zero stress

0

1000

2000

3000

4000

-1.5 -1 -0.5 0 0.5 1 1.5

33 )

E3 (MV/m)

a) Axial strain response

Saturated

Model

-2000

-1500

-1000

-500

0

-1.5 -1 -0.5 0 0.5 1 1.5

11 )

E3 (MV/m)

b) Transverse strain response

Saturated

Model

30

Figure 4 Hysteresis polarization responses under constant compressive stresses

31

Figure 5 Butterfly strain responses under constant compressive stresses

32

Figure 6 Hysteresis polarization responses under constant compressive stresses above the

coercive stress

33

Figure 7 Butterfly strain responses under constant compressive stresses above the coercive stress

34

Figure 8 Electro-mechanical responses under r=20/6 MPa/sec

-80

-60

-40

-20

0

-1.5 -1 -0.5 0 0.5 1 1.5

33 (MPa)

E3 (MV/m) a) Axial stress and electric field inputs

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

D3 (C/m2)

E3 (MV/m)

b) Electric displacement

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

-1.5 -1 -0.5 0 0.5 1 1.5

33 (%)

E3 (MV/m)

c) Axial strain

35

Figure 9 Electro-mechanical responses under r=80/6 MPa/sec

-80

-60

-40

-20

0

-1.5 -1 -0.5 0 0.5 1 1.5

33 (MPa)

E3 (MV/m) a) Axial stress and electric field inputs

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

D3 (C/m2)

E3 (MV/m)

b) Electric displacement

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

-1.5 -1 -0.5 0 0.5 1 1.5

33 (%)

E3 (MV/m)

c) Axial strain

36

Figure 10 The effect of frequency on the electro-mechanical coupling performance

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

D3 (C/m2)

E3 (MV/m)

a) Polarization f=0.5 Hz 0

0.02

0.04

0.06

0.08

-1.5 -1 -0.5 0 0.5 1 1.5

33 (%)

E3 (MV/m)

b) Strain f=0.5 Hz

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

D3 (C/m2)

E3 (MV/m)

c) Polarization f=1 Hz 0

0.02

0.04

0.06

0.08

-1.5 -1 -0.5 0 0.5 1 1.5

33 (%)

E3 (MV/m)

d) Strain f=1 Hz

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

D3 (C/m2)

E3 (MV/m)

e) Polarization f=10 Hz 0

0.02

0.04

0.06

0.08

-1.5 -1 -0.5 0 0.5 1 1.5

33 (%)

E3 (MV/m)

f) Strain f=10 Hz

37

Figure 11 Active composite plate with PZT patches

Figure 12 Tip deflection of the cantilever plate due to uniform cyclic electric fields applied to the

piezoelectric patches

38

Figure 13 The corresponding deformed shape of an active cantilever plate due to a uniform

cyclic electric field applied to the piezoelectric patches

Figure 14 Tip deflection of the cantilever plate due to a non-uniform cyclic electric field applied

to the piezoelectric patches

Highlights

We formulate a rate-dependent electro-mechanical coupling constitutive model.

The model can capture the hysteretic polarization switching and butterfly strain.

We present a numerical algorithm for electro-mechanical structural analyses.

Magnitude and rate of the electric field should be inputted to obtain shape

changes.