Thermodynamical modeling of ferroelectric polycrystalline ... · PDF fileThermodynamical modeling of ferroelectric polycrystalline material behavior ... such as PZT or Barium Titanate

Thermodynamical modeling of ferroelectric polycrystalline

material behavior

VOLKMAR MEHLING, CHARALAMPOS TSAKMAKIS, DIETMAR GROSSDarmstadt University of Technology

Department of Civil EngineeringHochschulstr. 1, 64289 Darmstadt

[email protected]

Abstract: A fully three-dimensional, thermodynamically consistent model for the behavior of polycrys-talline ferroelectric ceramics, such as PZT or Barium Titanate is presented. In an internal variableframework, the internal state of the material is described by two microscopically motivated internal statevariables. The first one is a second-order texture tensor, indicating the orientation of axes of the crys-tal unit cells. The second is vector-valued and describes the macroscopic polarization state. Similar torate-independent plasticity theory for small deformations, strains and electric displacements are decom-posed into reversible and irreversible parts. For the reversible part, a linear piezoelectric constitutivelaw is assumed. For the irreversible part, driving forces and evolution laws are established. Saturationand electro-mechanical coupling during irreversible processes are governed by energy barrier functionsintroduced in the electric enthalpy function. The model parameters are fitted to experimental hysteresisdata. Illustrative examples demonstrate the models capabilities.

Key–Words: Piezoelectricity, Electromechanical Coupling, Ferroelectricity, Thermodynamical Modeling

1 Introduction

Piezoelectric materials exhibit electromechani-cally coupled behavior. Due to the direct piezo-electric effect, the polarization is altered by ap-plied stresses, producing an electric signal. Theso-called inverse piezoelectric effect causes the de-formation of the material in an electric field. Botheffects are used in various applications for sensor-ing and actuation. For small applied stresses andelectric fields, the behavior of piezoelectrics is usu-ally considered as reversible. Piezoelectric mate-rials with the capability of repolarization underhigh electrical or mechanical fields are called fer-roelectrics. The group of ferroelectrics, this paperis concerned with, exhibits the tetragonal phase ofthe perovskite crystal structure. The best knownferroelectrics are BaTiO3 (Barium Titanate) andtetragonal PZT (Lead Zirconate Titanate). Thelatter is most frequently used for industrial ap-plication. The microscopic reason for repolariza-tion is the switching from one state to another ofunit cells or of domains with uniform orientationwithin the crystal . The domain switching processis dissipative such that the macroscopic behavioris hysteretic. For experimental results we referto [10, 3, 13, 4]. The number of publications ad-

dressing the simulation of uni-axial behavior offerroelectric materials is considerable. The de-velopment of reliable three-dimensional models isstill subject to intensive research. For an overviewof the literature the reader is referred to the re-view articles by Kamlah [6] and Landis [9]. Thebasic features of the model discussed in this paperhave been presented in [11]. The current paper isintended to present the state of current research.The model partly relies upon previous works byKamlah&Jiang [7] and Landis [8]. Analogousto the former, a set of internal state variables ischosen, which is motivated by microscopical con-siderations. Thermodynamical considerations areused to derive driving forces for the internal vari-ables. Similarly to the proceeding in incremen-tal plasticity theory, a loading function is used todivide reversible (piezoelectric) from irreversible(ferroelectric) processes. For the former, a lin-ear piezoelectric relation is assumed, applying aninvariant formulation cf. [12] including historydependent development of anisotropy for the me-chanical, the electrical as well as for the piezoelec-tric coupling properties. For the latter, normalityrules are used to determine the rates of change ofinternal state variables. Energy barrier functions

Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics, Chalkida, Greece, May 11-13, 2006 (pp191-196)

are used to model the saturation of irreversiblestrain and polarization, the saturation states be-ing defined by the boundaries of the admissiblerange of the internal state variables.

2 Notation

When index notation is applied, it refers to acartesian frame of reference and the indices are de-noted by small, slanted letters. Use is made of thesummation convention. Scalars are denoted byslanted letters (H,α,...), vectors are marked by ar-

rows (~P, ~n,...), while second, third and fourth or-der tensors are represented by bold face (A,σ,...),small double- stroke (dl, e,...) and capital double-stroke letters (C,P,...), respectively. The rules

~a·~b=aibi, tr (A)=Akk, (a·~b)ij =aijkbk, (a :B)k =

akijBij , (A·B)ij =AikBkj , (AT)ij =Aji, A :B=

tr(A·BT), (aT)ijk = akij and (C:A)ij = CijklAkl

hold, while |~a|=√~a·~a is the Euclidean norm of

the vector ~a and devA is the deviator of A. Wedenote by ~a⊗~b the tensorial product of two vectors

~a and ~b. Thus, e.g. ~a⊗B represents a third-ordertensor. Let A be a function of B. For the partialderivative with respect to B, we write ∂BA, while˙(·) is the material time derivative of (·). The di-

vergence and gradient operators are div (·) andgrad (·).

3 Model Summary

3.1 Thermodynamical Considerations

We start from the balance equations for mass, lin-ear and angular momentum, energy, entropy, elec-tric charge and magnetic flux. Neglecting electricbody forces and couples [6], as well as mechan-ical body forces, the balance equations of linearand angular momentum reduce to divσ = 0 ,where σ is the symmetric Cauchy stress ten-sor. We restrict ourselves to quasi-electrostaticprocesses and assume perfectly insulating prop-erties. Then, in the absence of external charges,

the Maxwell-relations reduce to div ~D=0, andthere exists an electric potential function ϕ, such

that ~E=− gradϕ . Here, ~D and ~E denote the elec-tric displacement and the electric field strength,respectively. The above field equations are sup-plemented by the Neumann-boundary conditions

σ · ~n = ~t and ~D · ~n = qf on surfaces withprescribed tractions ~t or prescribed free chargedensities qf and with surface normals ~n, and bythe Dirichlet-conditions for surfaces with pre-scribed displacements ~u or electric potentials ϕ.

Assuming, that the entropy flux and source termsare linked to the heat flux and radiation by the ab-solute temperature, and demanding the entropyproduction to be positive for all processes, onearrives at the Clausius-Duhem inequality

ψ+θs−ε·σ−pe+1

θ~q· grad θ = −πs ≤ 0, (1)

where ψ is the Helmholtz free energy, denotesthe mass density, θ and s are the absolute tem-perature and the entropy, respectively. ε and σ

denote the strain rate and Cauchy stress, pe isthe electrical power, ~q is the heat flux vector andπs is the entropy production. Now, the followingassumptions are made:◦ We consider isothermal processes with small de-

formations and uniform distribution of temper-ature.

◦ The material properties are assumed as time-independent.

◦ The electric enthalpy H, which is introduced bya Legendre transform

H = ψ − ~E · ~D − ~E · ~D, (2)

is a function of the electric Field ~E, the strain ε

and a set of internal state variables q. Further-more it is assumed to be additively decompos-able into two parts, Hr and H i, where H i onlydepends on q

H = H(ε, ~E,q) = Hr +H i (3)

Hr = Hr(ε, ~E,q) , H i = H i(q) . (4)

With this, equation (1) can be rewritten in termsof H as

(

σ − ∂εH)

: ε −(

~D + ∂~EH

)

· ~E

−∑

j

∂qjHqj ≥ 0 . (5)

Applying standard arguments of thermodynam-ics, it follows, that

σ = ∂εHr and ~D = −∂~E

Hr , (6)

together with the dissipation inequality

D :=∑

j

(−∂qjH)qj ≥ 0 (7)

are sufficient conditions for the validity of the sec-ond law of thermodynamics in the form of theClausius–Duhem inequality for every admissi-ble process.


3.2 Internal State Variables

The following additive decompositions into re-versible and irreversible parts of the strain tensor

ε and the electric displacement ~D are assumed toapply (cf. e.g. [2]):

ε = εr + ε

i , ~D = ~Dr + ~Pi . (8)

Here it is assumed, that εi and ~Pi are functions

of q (cf. [7]):

εi = ε

i(qj), ~Pi = ~Pi(qj) . (9)

For the polycrystalline ferroelectrics with tetrag-onal crystal structure considered here, the crystalunit cells can be characterized by the orientationof their longer crystal axis, in the following re-ferred to as ’c-axis’ and by the direction of thespontaneous polarization of that cell. Note, thata cell with a given c-axis may be polarized onlyparallel to this axis. In particular, with~c denotingthe direction of the c-axis, the polarization can be±Ps~c, where the material constant Ps is the mag-nitude of spontaneous polarization. We define theorientation distribution function (ODF)

f =3

4π~c · A ·~c , (10)

where A is referred to as the texture tensor ofthe distribution and denotes a symmetric, positivesemi-definite second-order tensor, with tr (A) =1. Function f can be interpreted as a surface-density function on the unit sphere. Then, inte-gration over all directions ~c yields

∫∫

f dA = 1.The irreversible strain state of a unit cell is

directly connected to the direction of its c-axis.In [11], volume averaging over all unit cells is usedto derive the macroscopic irreversible strain

εi =

3

2ε0

(

A − 1

31

)

. (11)

The texture tensor A serves as an internal statevariable, that describes the irreversible strainstate. An irreversible strain state is consideredas saturated, if one or two eigenvalues of A equalzero. For the representation of the macroscopicremanent polarization, a vector-valued state vari-able ~p is chosen, such that

~Pi = P0~p . (12)

ε0 and P0 are the saturation values of uniaxialstrain and polarization of the polycrystal. The

extent to which the material can be polarized de-pends on the orientation of c-axes. It has beenshown in [11], that the texture-dependent limitstates (saturation states) of polarization in a givendirection ~n, |~n| = 1 can be expressed by the po-larizability function

~Pisat = P0

(

A · ~n +1

2(1 − ~n · A · ~n)~n

)

. (13)

This function is comparable to the ’distance vari-able’ applied in [8]. For simplicity, (13) is approx-imated by an ellipsoidal form

~Pi ⋆sat =

1

2P0 (1 + A) · ~n , |~n| = 1 . (14)

The admissible range for ~p can be written as

{

~p |2(1 + A)−1 · ~p| ≤ 1}

, (15)

and the distance of the relative polarization fromthe saturation surface can be described by thevariable

η = 2|(1 + A)−1 · ~p| . (16)

It is zero for vanishing polarization and equal toone in case of polarization saturation.

3.3 Constitutive Law

When leaving the switching processes out of con-sideration, the behavior of ferroelectrics is purelypiezoelectric. It is common, to assume the piezo-electric relation between the mechanical and elec-trical state variables to be linear (e.g. [5]). Inour case the piezoelectric part of the electric en-thalpy function is formulated by application of aninvariant scheme. It can be written in the follow-ing quadratic form

Hr = Hr(εr(ε,A), ~E, ~p) (17)

=1

2ε

r:CE(~p):εr − εr:e(~p)·~E − 1

2~E·ǫε(~p)·~E − ~Pi·~E ,

with the tensors of elastic, piezoelectric and di-electric moduli C

E, e and ǫε, respectively, being

functions of the polarization state of the poly-crystal. In this sense, the piezoelectric relationis assumed to be transversely isotropic, with theaxis of anisotropy aligned with the vector of irre-versible macroscopic polarization. For vanishingmacroscopic polarization (i.e. |~p| = ~0) all modulibecome isotropic and piezoelectric coupling dis-appears. This seems reasonable for the thermallydepolarized state of the polycrystal.


Similarly to [7] and [8] the part of the electricenthalpy function, which is concerned with irre-versible processes is composed of quadratic func-tions, and singular terms, which tend to infinityas internal variables approach saturation states:

H i = H i(A, ~p) (18)

=1

2cAtr dev (A)2 +

1

2cp~p · ~p

+aA

mA

tr (A−mA) +ap

mp(1 − η)−mp. (19)

The third term on the right hand side of (18) rep-resents an energy barrier function, which accountsfor strain saturation. It grows to infinity as anyeigenvalue of the texture tensor approaches zero.Analogously, the last term tends to infinity as thegap between the current amount of polarizationand the polarizability vanishes. cA, aA, cp, ap, mA

and mp are material parameters.Using the above definitions, we can identify

the terms in the brackets of equation (7) as driv-

ing forces fA and ~fp for the internal variables A

and ~p.

fA = −∂AH i ,~fp = −∂~pH

r − ∂~pHi.

(20)

Well motivated by the experimental observa-tion, namely that irreversible processes only oc-cur in case of loading beyond certain limits, a so-called switching function is introduced to distin-guish reversible from irreversible processes. It iscompletely analogous to yield functions in plastic-ity theory. In particular, the following switchingfunction is used in the examples below,

F =dev (fA)

fAc

:dev (fA)

fAc

+dev (fA)

fAc

:(~p⊗~fp

fpc

)sym

+|~fp|2(fp

c )2− 1 ,

(21)where fA

c =√

1.5ε0σc and fpc = P0Ec represent

critical driving forces in uniaxial experiments. Inthese equations, Ec and σc are the coercive elec-tric field and the coercive stress, respectively. Thepostulation of maximum electro-mechanical dis-sipation during switching processes leads to thenormality rules for the evolution of the internalstate variables

A = λ∂fAF , ~p = λ∂~fpF . (22)

The factor λ is determined from the so-called con-sistency condition F = 0.

-2 -1 0 1 2-0.001

0.000

0.001

0.002

0.003

0.004

-2 -1 0 1 2-0.5

-0.25

0

0.25

0.5

E [kV/mm] E [kV/mm]

εD[ µC

mm2]

Figure 1: Experimentally observed (dashed lines)and simulated (solid lines) D-E (left) and ε-E(right) hysteresis loops.

4 Numerical Simulations

The model has been implemented in a finite el-ement code to facilitate the simulation of com-plex non-homogeneous setups. The calcula-tions have been carried out using the MatLab

based open-source finite element code DAEdalon(www.daedalon.org), employing three dis-placement components and the electric potentialas nodal degrees of freedom. Time-integration ofthe rate-equations is carried out by the backward-Euler scheme together with the predictor-corrector method known from computationalplasticity.

The parameters of the model have beenroughly fitted to experimental hysteresis datafrom uniaxial electric cycling [1]. The measuredand simulated hysteresis loops are depicted inFig.1. The agreement of both sets of graphs isvery good, though some discrepancy remains inthe strain hysteresis. It is clear, however, thatadditional effort and more data, e.g. from purelymechanical experiments, are required, to fit allparameters in a satisfying way.

In Fig.2, the model behavior for differenttypes of uniaxial loading is shown. All of theseare in good qualitative agreement with experi-mental observations reported e.g. in [3, 10, 13].Figs.2a and b show the model response to elec-tric cyclic loading with different superimposedconstant mechanical stresses. Compared to thecase without the mechanical loading, one can ob-serve, that small compressive stresses (-15MPa)increase the ratio of maximum to minimum strain,while high stresses inhibit the evaluation of tex-ture and thus lead to degenerated butterfly hys-teresis loops. Accordingly, the electric hystere-sis loop for high compressive stress is reduced tothe part which is related to pure 180◦ switch-ing (change of polarization without change of tex-ture). During mechanical depolarization (Fig.2cand d), a previously polarized ferroelectric is sub-


a) b)

-2 -1 0 1 2

-0.25

0

0.25

-2 -1 0 1 2-0.002

0.000

0.002

0.004

0.006

E [kV/mm] E [kV/mm]

εD

[ µC

mm2]

σ [MPa]

600-15-60

c) d)

-0.002 0 0.002-150

-100

-50

0

50

-0.3-0.2-0.1-150

-100

-50

0

50

D [µC/mm2] ε

σ[MPa]

σ[MPa]

Figure 2: Model response a) D − E-hysteresisloops for different magnitudes of superimposedconstant stress. b) butterfly hysteresis connectedwith a). c) total strain during depolarization bycompressive stress. d) polarization during depo-larization.

jected to compressive stress along the axis of orig-inal poling. Thereby, the material is partly de-polarized, accompanied by a negative irreversiblestrain.

The graphs in Fig.3 resemble the multiax-ial polarization rotation experiment reported in[4]. Here, a previously poled ferroelectric is sub-jected to an electric field, which differs fromthe original poling direction by different angles

a) b)

-4-3-2-10

-1

-0.75

-0.5

-0.25

0 -4-3-2-10-0.003

0.000

0.003

0.005

180◦

135◦90◦

45◦

0◦

c) ~E

180◦

135◦

90◦

45◦

0◦

E [kV/mm] E [kV/mm]

D[ µC

mm2] ε 90◦

45◦

135◦

180◦

0◦

~p

Figure 3: Polarization rotation: electric loading,with the original poling direction differing by α

from the direction of the applied electric field. a)electric response, b) strain response, c) evolutionof relative polarization.

a)

∆ϕ

Air

PZTB

C

D l

b)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

|~p|[-]

A B C

D

c)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

2

|~E|[ kV

mm]D

A B C

d)

0 5 10 150

5

10

15

00.250.50.7511.251.51.752

E[ kV

mm]

A B x [mm] C

ϕ[kV]

Figure 4: a) Schematics of the simulation of aquarter of a strip with a circular hole. The po-tential boundary conditions are applied to theleft and right boundaries. b) Contour plot ofthe magnitude of relative irreversible polarization|~p|. Maximum value is 0.78 at point D. c) Con-tour plot of the magnitude of the electric field |~E|,Maximum value inside the hole is 1.77 kV/mm atpoint B, in the ferroelectric 1.38 kV/mm at pointD. d) Electric potential ϕ (solid line) and elec-tric field strength E (dotted line) along the lineof symmetry A-B-C at an average electric field of1kV/mm.

α = 0◦, 45◦, 90◦, 135◦ and 180◦. The responsestrongly depends on α. The change in electricdisplacement D is increasing with increasing an-


gles, the change in strain is largest for α = 90◦.For α = 180◦, the results are exactly one half ofthe hysteresis loops in Fig.1. The small graphin Fig.3c depicts the evolution of relative polar-ization, starting from the different initial states,indicated by the arrows. The applied electric fieldis pointing to the left.

The aim of the last example (Fig.4) is todemonstrate the capability of the model to predictcomplex inhomogeneous loading states. An ini-tially unpolarized ferroelectric strip (PZT) with acircular hole (35×20mm) has electrodes attachedto its short ends. The hole is modelled as di-electric (air/vacuum). To reduce the calculationeffort, only one quarter of the system is disce-tized (see Fig.4a). The potential difference ∆ϕ isincreased linearly from zero to 17.5 kV (for thequarter system). Thus, the average electric fieldstrength at the final state of loading is 1 kV/mm,which is slightly above the coercive field of 0.9kV/mm. It becomes clear, that the electric fieldinside of the hole is much higher than the one inthe ferroelectric. For comparison the same calcu-lations have been carried out without discretiza-tion the hole, i.e. with a free surface on the in-terface. The results are almost identical to theones presented in this paper. This indicates, thatthere is no need to account for the dielectric prop-erties of the air in the hole. However, one shouldkeep in mind, that high electric fields could leaddo electric discharge in an experimental setup.

5 Conclusion

A constitutive model has been presented, whichaccounts for multi-axial switching and its capabil-ities to reproduce experimental results have beendemonstrated. The more elaborated fitting of allconstitutive parameters remains a difficult and la-borious task and is beyond the scope of this work.The restriction to materials with tetragonal struc-ture is convenient. However, the proceeding couldbe generalized for other or additional structures.

Acknowledgements: We thank N. Balke andJ. Rodel of the Department of Material Sciencesat the Darmstadt University of Technology forproviding experimental data.

References:

[1] N. Balke, D.C. Lupascu, T. Granzow andJ.Rodel. Unipolar fatigue and dc loadingof lead zirconate titanate ceramics. Acta.Mater, submitted. 2006.

[2] E. Bassiouny, A.F. Ghaleb and G.A. Mau-gin. Thermomechanical formulation for cou-pled electromechanical hysteresis effects – I.Basic equations, II. Poling of ceramics. Int.J. of Eng. Sci., 26,1279–1306, 1988.

[3] T. Fett, S. Muller, D. Munz and G. Thun.Nonsymmetry in the deformation behaviourof PZT. J. of Mater. Sci. Lett., 17,261–265,1998.

[4] J.E. Huber and N.A. Fleck. Multi-axial elec-tric switching of a ferroelectric: theory versusexperiment. J. Mech. Phys. Sol., 49,785–811,2001.

[5] B. Jaffe, W. R. Cook and H. Jaffe. Piezo-electric Ceramics, Academic Press, London,1964.

[6] M. Kamlah. Ferroelectric and ferroelasticpiezoceramics - modeling of electromechani-cal hysteresis phenomena. Cont. Mech. Ther-modyn., 13(4),219–268, 2001.

[7] M. Kamlah and Q. Jiang. A constitutivemodel for ferroelectric PZT ceramics un-der uniaxial loading. Smart Mater. Struct.,8,441–459, 1999.

[8] C.M. Landis. Fully coupled, multi-axial,symmetric constitutive laws for polycrys-talline ferroelectric ceramics. J. Mech. Phys.Sol., 50,127–152, 2002.

[9] C.M. Landis. Non-linear constitutive model-ing of ferroelectrics. Curr. Op. Sol. St. Mater.Sci., 8,59–69, 2004.

[10] C.S. Lynch. The effect of uniaxial stress onthe electro-mechanical response of 8/65/35PLZT. Acta mater., 44(10),4137–4148, 1996.

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Thermodynamical modeling of ferroelectric polycrystalline ... · PDF fileThermodynamical modeling of ferroelectric polycrystalline material behavior ... such as PZT or Barium Titanate