Multiaxial behavior of ferroelectric single and polycrystals with pressure dependent boundary effects

Computational Materials Science 44 (2009) 1222–1230

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Multiaxial behavior of ferroelectric single and polycrystals with pressure dependent boundary effects K. Jayabal, A. Arockiarajan, S.M. Sivakumar * Department of Applied Mechanics, Indian Institute of Technology Madras, 600036 Chennai, India

a r t i c l e

i n f o

Article history: Received 7 June 2008 Received in revised form 15 July 2008 Accepted 11 August 2008 Available online 17 September 2008 PACS: 77.90+K 77.80.Fm 87.80.Ek Keywords: Ferroelectrics Domain switching Micromechanical modeling Boundary effects Multiaxial loading

a b s t r a c t The external electric and mechanical fields applied at angles to the initial poled direction of the ferroelectric ceramics produce a significantly different nonlinear behavior to that of external fields applied parallel to the poling direction. This angle dependent response of ferroelectric single and polycrystals are predicted by the model proposed based on irreversible thermodynamics and physics of domain switching. The dissipation associated with boundary constraints in thin ferroelectric single crystals are incorporated in the model. As well, the pressure dependent constraints imposed by the surrounding grains on the grain of interest at its boundary during domain switching is correlated with the resistance experienced by a ferroelectric single crystal on its boundary during domain switching. Taking all the domain switching possibilities, the volume fractions of each of the variants in a grain are tracked and homogenized for macroscopic behavior. Numerical simulations were carried out for the multiaxial behavior of ferroelectric single and polycrystals under electrical, mechanical and electromechanical loading conditions. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Ferroelectrics are employed in sensors and actuators in the piezoelectric form due to their strong electromechanical coupling behavior. Typically, the ferroelectric ceramics operate only within their linear region under smaller fields, however, they exhibit appreciable nonlinearity under higher electric and mechanical fields owing to underlying microscopic domain switching. The ferroelectric ceramics are generally positioned in the devices in such a way that they are subjected to uniaxial loadings in order to have the maximized utilization of the piezoelectric coupling for the actuation and sensing purposes. When subjected to even uniaxial loadings, under a critical combination of the electrical and mechanical fields, the ferroelectrics tend to change their macroscopic remanent polarization due to the microscopic domain switching. Since this will alter the piezoelectric coupling coefficients of the ferroelectrics, the devices employed them may lead to unreliable results. Hence, the behavior of the ferroelectric ceramics under complex uniaxial loading cases is essential before they become part of the devices. Realizing this, many experiments were conducted on them under uniaxial loading conditions where * Corresponding author. Tel.: +91 44 9513874; fax:+91 44 9513898. E-mail address: [email protected] (S.M. Sivakumar). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.08.004

the external fields are applied parallel to the initial poled direction of the crystals [1–4]. The ferroelectric constitutive models are calibrated and validated against the uniaxial experimental data. However, the uniaxial experiments, in spite of the different combination of electrical and mechanical fields, being simple in nature will not suffice to assess the models completely. The multiaxial experiments are complex in nature and will provide the additional details required to validate the models in a wider perspective. Also, the ferroelectrics employed in the present day devices are no longer expected to undergo only uniaxial loadings due to their complex geometries upon demand from the device designers. In addition, the electrode tips and the inhomogeneities present in the ferroelectrics act as field intensifiers forcing the external fields to act at angles to the initial poled direction. These kind of multiaxial loadings will make the nonlinear response more complicated resulting in the ferroelectric devices performing differently to that of forecast under uniaxial loading cases. Hence, the knowledge in the multiaxial response of the ferroelectric ceramics assumes significance in order to enhance the predicting capability of the devices considering the severe environments they are put in use. In view of this, a simple version of the multiaxial electrical switching of ferroelectric ceramics (PZT-5H) was studied in [5–7] where the electrical fields were applied with certain inclinations

K. Jayabal et al. / Computational Materials Science 44 (2009) 1222–1230

to the initial remanent polarization direction. Multiaxial ferroelastic switching using tubular specimens with internal and external pressures, and axial loads was analyzed in [8]. These multiaxial experiments were carried out to construct the yield surfaces for comparison with the model predictions to understand the applicability and the limitations of the models. In the multiaxial loading experiments conducted in [5,7], the ferroelectrics were initially poled in a specific direction followed by the application of electric fields at angles to the initial poled direction. The outcome of one such experiment in terms of the change in the remanent strain and polarization measured along the direction of the electric field is reproduced from [7] in Fig. 1. The dependence of the nonlinear response on the initial polarization orientation becomes apparent from the variation in the behavior of the ferroelectric ceramics with respect to the angle of rotations. The multiaxial experiments on the ferroelectric ceramics were so far carried out or reported in the literature to our knowledge either with electric or with mechanical fields. A combined electromechanical multiaxial loading may provide additional information for better assessment of the models. In the actuation devices, the ferroelectric polycrystals are employed in the stacked form to enhance the actuation strain as the levels of strain provided by individual crystals are limited. Several efforts have been directed toward achieving a large actuation and one possible solution was demonstrated using the relaxor single crystals [9,10]. But, those crystals require precise composition control and need very high electric fields for producing large strains. An alternative solution for generating large actuation strains could be thorough the utilization of the domain switching in ferroelectric single crystals [11]. The large actuation strain achieved in the ferroelectric single crystals using cyclic electric field was primarily due to the constant compressive stress applied on the boundary of the crystal. The friction between the crystal boundary and the loading platen was believed to play an appreciable role in the response of the thin ferroelectric single crystals [12]. Considering the potential use for the actuation purposes, the multiaxial behavior of the ferroelectric single crystals inclusive of the boundary effects assumes importance from the design point of view. In line with that, an attempt is made in this work to capture the multiaxial response of the ferroelectric single crystals under electrical, mechanical and electromechanical loading conditions. Ferroelectric constitutive models can be classified in general as phenomenological models and micromechanical models and a detailed review of them is reported in [13–15]. An early domain switching micromechanical model [1] makes an assumption that the domain switching occurs instantly upon the domains reaching the critical energy state. Modifications have since been proposed to simplify or to broaden the range of applicability of these models

1223

[16,3]. These models assume a sudden switching unlike what happens in the realistic behavior of ferroelectric single crystals under combined electromechanical loading conditions. Also, the effects due to the interaction of neighboring grains were ignored to reduce the computational effort. Some efforts towards including the interaction effects into the domain switching models have been attempted using either modification of the loads [17,18] or using self consistent homogenization methods [19]. An attempt was made to include the grain boundary effects in the micromechanical model via a phenomenologically motivated probabilistic approach [20]. However, the computational cost of the self consistent models and the finite element based micromechanical models is very high. Retaining the advantage of the micromechanical modeling and at the same time to preserve the computational cost, a simple micromechanical model was proposed to include the boundary effects by modifying the switching criterion appropriately [21]. In this approach, the resistance encountered by a grain due to its surrounding grains in a ferroelectric polycrystal under the influence of stress was correlated with the dissipation experienced by a constrained thin ferroelectric single crystal during domain switching. The model based on this switching criterion was found to capture well the uniaxial response of the ferroelectric polycrystals and single crystals with constrained boundary. This paper is arranged as follows: In the next section, the model used is briefly described with a view to introduce the significance of the inclusion of the boundary effects. Simulations carried out for the behavior of the single crystals and polycrystals under multiaxial loads is described in the subsequent section. The paper ends with salient concluding remarks.

2. Model formulation 2.1. Driving force Ferroelectric ceramics undergo a phase transformation from a higher symmetry paraelectric phase to a lower symmetry ferroelectric phase due to charge center separation at the unit cell level on its temperature falling below Curie temperature. A spontaneous polarization and an associated strain is realized in the ferroelectric phase with reference to paraelectric cubic structure. In the ferroelectric phase, a typical unit cell may possess different crystal structures like tetragonal, rhombohedral or a combination of them depending upon the constituents of the ceramics. Consideration of the tetragonal crystal structure for the entire material in the ferroelectric phase is observed to yield reliable results in the literature. The macroscopic response of the ferroelectrics can be obtained by averaging the microscopic behavior of the unit cells. A ferroelectric

Fig. 1. The dependence of the nonlinear response of the ferroelectric polycrystals on polarization rotation [7].

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ceramic with a tetragonal crystal structure can have six distinct domains (crystal variants) separated by domain walls. The presence of each crystal variant can be represented by their volume fractions, fi. The externally applied electrical, mechanical or combined loading may prefer a specific variant at the expense of other variants and the volume fraction of each of them at any instant of loading must satisfy ðnÞ

f

X

P 0;

f

ðnÞ

¼ 1;

X

f_ ðnÞ ¼ 0;

n ¼ 1; 2; . . . 6:

L ij

N ij ;

eij ¼ e þ e

ð2Þ

PLi

PNi :

ð3Þ

þ

A ferroelectric polycrystal contains many grains and each of them contain a unique or several domains depending upon the loading history it had undergone. The macroscopic behavior of the ferroelectrics can be approximated by averaging the microscopic domain response within each grain for the given loading. Hence, from Eqs. (2) and (3), the total strain and polarization of the material is realized by adding the reversible and irreversible parts of individual domains as

eij ¼

6 X

ðnÞ

fðnÞ eij ¼

n¼1

Pi ¼

6 X

6 X

LðnÞ

fðnÞ ðeij

NðnÞ

þ eij

Þ;

ð4Þ

n¼1 ðnÞ

fðnÞ Pi

¼

n¼1

6 X

LðnÞ

fðnÞ ðPi

NðnÞ

þ Pi

Þ;

ð5Þ

n¼1 NðnÞ ij

NðnÞ Pi

where, e and represent the sponataneous strain and polarization of the variant, n, respectively. As well, the Gibb’s free energy density of the entire crystal, neglecting grain to grain interactions, can be obtained by the sum of contributions from the individual domains

gðrLij ; ELi ; T; fÞ ¼

6 X

fðnÞ g ðnÞ :

ð6Þ

n¼1

The Gibb’s free energy density can be related to internal energy density u through

qg ¼ qu  r

L ij ij

e 

Ei PLi

 qT g:

ð7Þ

Remembering the second law of thermodynamics and using Eqs. (1)–(7), the dissipation inequality can be derived as

" 0 6  qg ;rij þ

6 X

#

"

eLðnÞ r_ ij  qg ;Ei þ ij

6 X

n¼1

þ

6 h X

# LðnÞ

Pi

E_ i

n¼1

i

NðnÞ rij e_ NðnÞ þ Ei P_ i  f_ ðnÞ qg ;fðnÞ : ij

ð8Þ

n¼1

The variation in the total strain and polarization of the crystal can be determined by averaging their corresponding values from the individual variants which can be obtained through the volume fractions

e_ NðnÞ ¼ ij

6 X

DeNðm!nÞ f_ ðm!nÞ ; ij

for m–n;

ð9Þ

DPNðm!nÞ f_ ðm!nÞ i

for m–n;

ð10Þ

m¼1 NðnÞ P_ i ¼

6 X m¼1

06

Nðm!nÞ

6 X 6 X

Nðm!nÞ

½rij Deij

Nðm!nÞ

þ E i DP i

 qDg ðm!nÞ f_ ðm!nÞ

for m–n

n¼1 m¼1

ð1Þ

Ferroelectric ceramics exhibit reversible response under smaller external fields and demonstrate irreversible changes for sufficiently higher fields. The response of the ferroelectrics for higher external loads in terms of the total strain and polarization at a given instant can be additively decomposed as reversible (almost linear, L) and irreversible (nonlinear, N) strain and polarization, respectively,

Pi ¼

Nðm!nÞ

where Deij and DPi indicate the change in the spontaneous strain and polarization, respectively, pertain to domain switching. Here, D()(m ? n) = ()(n)  ()(m). For mechanical and electrical equilibrium conditions, the controllable variables disappear and Eq. (8) turns out to be

ð11Þ (m ? n)

where Dg corresponds to the change in the Gibb’s energy density concerned to the domain switching and can be derived as

qDg ðm!nÞ ¼ 

i 1h ðm!nÞ rij DSðm!nÞ rkl þ Ei Dvðm!nÞ Ej þ 2rij Ddkij Ek : ij ijkl 2 ð12Þ

The change in the Gibb’s energy associated with the domain switching is due to the variation in the material properties of the domains involved in the switching process. When the variation in the elastic stiffness Sijkl and the dielectric permittivity vij may be neglected between the domains involved in the switching process, the difference in the piezoelectric coupling coefficient dijk can not be ignored as the variation is significant [22,23]. Hence, the required dissipation inequality is obtained as

06

6 X 6  X



ðm!nÞ Nðm!nÞ rij DeNðm!nÞ þ E i DP i þ rij Ddkij Ek f_ ðm!nÞ ij

n¼1 m¼1

for m–n 06

6 X 6 X

ð13Þ ðm!nÞ

ðfdrive Þf_ ðm!nÞ

for m–n:

ð14Þ

n¼1 m¼1

The above inequality is valid subject to the constraints given in Eq. (1). The first two terms in the driving force appearing in Eq. (13) concern the external stress and electric fields while the third term concerns the driving forces attributed to the domain to domain ðm!nÞ interactions. When the driving force, fdrive , transforming a variant from m to another variant n, reaches a critical value fc, the domain switching is initiated. If we refer the domain switching process involving two distinct crystal variants to a transformation system, then, there are as many as 30 transformation systems possible for a tetragonal crystal ferroelectrics (in this, m ? n switching is taken to be different from n ? m). For the transformation system considered above to be active, the driving force must satisfy the following inequality at all times: ðm!nÞ fdrive f_ ðm!nÞ 6 fc f_ ðm!nÞ

ð15Þ

It is assumed here that the transformation is stable, i.e. the rate of resistance to the transformation is higher than the rate of the driving force. When the domain switching takes place from variant m to variant n, it is referred to as one transformation system and the vice versa is referred to another transformation system. A priori, it will not be known which of the transformation system will be active. When the above switching takes place, both m to n, as well as n to m transformation systems will be active and the changes in volume fractions, f_ ðm!nÞ and f_ ðn!mÞ will be of same magnitude with opposite signs, positive and negative, respectively. Hence, the driving forces for both the transformation systems are determined but only the transformation system producing the positive volume fraction is considered and from that the diminishing volume fraction is obtained. This avoids a redundancy and duplication of changing the volume fractions of the two variants m and n in the algorithm. 2.2. Switching criterion under constrained boundary condition A switching criterion assumes significance in the micromechanical modeling as it determines the onset of domain switching for

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the given external fields. Several quantities such as the work done [1], the Gibbs free energy [3], the total potential [18] and the internal energy density [24] have been utilized in the switching criterion. The switching criterion used in this work is a refinement of the one earlier proposed [21]. The switching criterion derived for a specific loading such as electrical loading can be applied for any generalized combined loading conditions. When the 90° domain switching occurs, it can crystallographically be shown that the interface between the domains involved in the switching is 45°. Considering the fact that a domain can undertake any of the four 90° domain switching in a three dimensional perspective, there would be four possible interface planes identified in a unit cell. It was assumed that the 90° domain switching takes place as soon as the resolved external electric field along the interface reaches the coercive electric field. Thus, the critical value that should be attained by the driving force for the occurrence of 90° switching can be derived from Eq. (13) as

fc90 ¼

pffiffiffi 90 N 2Ec DP :

ð16Þ

As the underlying mechanism for the 90° domain switching is different from that of 180° domain switching, the critical values for their occurrence are also different. Also, the coercive electric 180 for initiating 90° and 180° domain switching, fields E90 c and Ec respectively, under pure electrical loading are considered to be different. When the applied electric field in the opposite direction of the polarization direction in a unit cell attains the coercive electric field E180 c , the 180° domain switching takes place. From Eq. (13), the critical value for the driving force to induce 180° switching can be obtained as N fc180 ¼ 2E180 c DP :

ð17Þ

It is to be noted that the critical value for the 180° switching has in fact been effected in many models in the literature originating and E90 from [1]. The coercive electric fields E180 c c can be obtained from the experiments. Though Eqs. (16) and (17) represent the critical values that should be met by the driving force to induce the domain switching for an electrical loading condition, it can be employed to any general electromechanical loading conditions. However, these critical values will not suffice for some special cases, for instance, where a significant boundary dissipation occurs as discussed below. A considerable improvement in the actuation strain was demonstrated by subjecting a thin ferroelectric single crystal (BaTiO3) to a cyclic electric field with constant compressive stress [25]. It was observed during the experiment that the actuation strain increased appreciably with the magnitude of the compressive stress. But, the measured actuation strain was not comparable with the theoretically derived strains. Shilo et al. [12] reasoned for the reduced actuation strain from the experiments as the resistance arising out of the friction between the loading platen and the crystal surface during the domain switching. Under combined electromechanical fields, the normal strain developed in the ferroelectric single crystal was accompanied by a lateral strain causing the crystal to elongate or contract laterally as the domain wall moves through the crystal. This lateral motion of the crystal was resisted by the friction between the crystal boundary and the loading platens and there is an associated energy dissipation. For a ferroelectric ceramic possessing the tetragonal crystal structure, the macroscopic dimension change is effected only by the microscopic 90° domain switching. This inference is based on fact that the 90° domain switching is associated with both the strain and polarization change whereas the 180° domain switching is related only to the polarization change. Hence, as the 90° domain wall moves in a constrained thin ferroelectric single crystal, it experiences a certain amount of additional resistance pertained to the dissipation between the crystal boundary and the loading platen which depends

on the coefficient of friction and the external stress applied on the boundary. This pressure dependent boundary dissipation is incorporated into the model through the switching criterion in a straight forward way

fc90 ¼

pffiffiffi 90 N 2Ec DP þ Kð< rii >ÞDf

ð18Þ

where Df represents the increment in the volume fraction of the favored domains for the load step increment. The constant K depends upon the friction between the crystal surface and the loading platen. The lower the friction, the less is the associated energy dissipation. In the absence of stress or under a tensile stress, the term expressed in Macaulay bracket in Eq. (18) disappears. As the strain contribution from the 180° domain switching is negligible, the 180° domain wall motion is assumed not to face any additional resistance. Hence, its critical value remains as derived in Eq. (17). It is important to mention here that the boundary dissipation is not realized by the crystal under pure electrical loading as the pressure on the crystal boundary is negligible. Eventually, the generalized switching criterion inclusive of the boundary effects for the 90° and 180° domain switching, respectively, can be expressed as,

pffiffiffi

ðm!nÞ Nðm!nÞ N rij DeNðm!nÞ þ E i DP i þ rij Ddkij Ek P 2E90 c DP ij þ Kð< rii >ÞDfðm!nÞ ; ðm!nÞ Nðm!nÞ N rij DeNðm!nÞ þ E i DP i þ rij Ddkij Ek P 2E180 c DP : ij

ð19Þ ð20Þ

Here, Df(m ? n) indicates the increment in the volume fraction of variant n transformed from variant m for the given load step. Nðm!nÞ Nðm!nÞ ¼ sij Deij where rij and sij represent In Eq. (19), rij Deij the total and the deviatoric part of the stress, respectively. The irreversible strain associated with the domain switching, DeNðm!nÞ , is deviatoric in nature. Hence, it may be noted that the ij deviatoric part of the stress takes part in the left hand side of Eq. (19) and the hydrostatic part of the stress (related to the trace of the external stress providing the normal component associated with the frictional dissipation) occupies the right hand side. The above switching criterion applicable for a constrained thin ferroelectric single crystal can also be extended to ferroelectric polycrystals. Polycrystal ceramics contain a large number of grains with different crystallographic orientations and each of them is separated from others through grain boundaries. Owing to the difference in the orientations of the grains, there is a mismatch in the strain and polarization in each grain across the grain boundaries. During external loading, the domain wall sweeps across the grain to produce a specific crystal variant that is favored among all possible variants. The switching criterion given in Eqs. (16) and (17) would suffice if all grains have the same crystallographic orientation. But, the existence of grain boundaries demands an additional driving force to overcome the resistance arising out of the strain and polarization mismatch at the boundaries. It is thought to be reasonable to draw an analogy between a constrained thin ferroelectric single crystal and a grain in a ferroelectric polycrystal constrained by the surrounding grains. As well, the dissipation on the grain boundary in a ferroelectric polycrystal under the influence of stress during domain wall motion may be comparable to that of pressure dependent boundary dissipation in a constrained ferroelectric single crystal during domain switching. Here, the dissipation concerning to the polarization mismatch on the grain boundary arising from 180° domain switching is ignored as was done for the single crystals to make the model simpler, yet found to predict the behavior of ferroelectric polycrystals reasonably. Thus, the generalized switching criterion derived in Eqs. (19) and (20) based on the constrained ferroelectric single crystal can also be employed for a grain in a ferroelectric polycrystal under electromechanical loading conditions. The constant K in Eq. (19) is to be determined by calibrating the model against the experimental

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K. Jayabal et al. / Computational Materials Science 44 (2009) 1222–1230

results. However, as the emphasize was laid on modeling only the influence of the pressure dependent boundary effects, it is to be noted here that the boundary effects in the absence of stress are not considered. 3. Numerical examples 3.1. Simulation procedures At a given instant, a ferroelectric crystal may possess many variants depending on the previous loading history it had undertaken. When the external fields are applied, the volume fractions of the variants in the crystal are altered as favored by the fields. To determine the evolution of the variants, the driving force of each transformation system is calculated from Eq. (13) and compared with the switching criterion given in Eqs. (19) and (20). When the driving force is lower than the critical values, the response of the crystal is reversible with the composition of the volume fractions of the variants in the crystal unaltered. On the driving force satisfying the 180° switching criterion in Eq. (20), the switching variant is entirely transformed into the favored variant. But, in the case of 90° domain switching, an incremental switching is set to happen as the driving force attains the critical value defined in Eq. (18). The increment in the favored variant Df is determined in a straight forward way from Eq. (19) such that the driving force becomes slightly less than the right side term. The changes in the volume fractions are updated for each variant to the given load step and the response of the crystal is obtained by

egij ¼

6 X ðnÞ ðnÞ NðnÞ ðSijkl rkl þ dkij Ek þ eij ÞfðnÞ ;

ð21Þ

n¼1

Pgi ¼

6  X

ðnÞ

NðnÞ

ðnÞ

dijk rjk þ vij Ej þ Pi

 fðnÞ ;

ð22Þ

n¼1

where, egij and P gi represent the total strain and polarization of the ðnÞ ðnÞ ðnÞ crystal, respectively, with Sijkl , dijk and vij indicating the material properties of the variant n. In the simulations, transversely isotropic properties were assigned to each variant and the switching process modified the domain properties according to the transformation law discussed with more details in [3,26]. In the single crystal simulations, one of the domain orientation directions was taken as the reference axis and the external fields were applied with respect to that reference axis. The ferroelectric polycrystal behavior can be obtained by averaging the responses of a number of single crystals and, here, 1000 crystals were found sufficient to generate the poly-

crystal behavior reasonably. Each crystal refers to a grain in the polycrystal. The orientations of the grains in the polycrystal were randomly selected based on Eulerian angles. A simple Reuss approximation, considering a uniform stress and electric field through out the material, was employed for averaging the crystal responses for polycrystal behavior. The material properties for ferroelectric single crystal BaTiO3, and for ferroelectric polycrystal PZT were considered from [25,12,3]. The value of the friction parameter K, a strong dependent on the friction conditions between the crystal surface and the loading platens, was assumed in the single and polycrystal simulations as 0.04 and 0.2, respectively. However, consideration of K as a function of stress may yield more agreeable results with the experiments. 3.2. Multiaxial response of single crystals The uniaxial response of a flat single crystal barium titanate was experimented by Burcsu et al. [25] under cyclic electric loads with and without constant compressive stresses. As the crystal undergoes almost a complete 180° domain switching in the absence of stress, no macroscopic strain was observed. However, the presence of the compressive stress on the crystal boundary induced more 90° domain switching resulting in an appreciable improvement in the actuation strain. The pressure dependent boundary resistance (frictional resistance) was reasoned for the experimentally observed strains to be considerably lower than the theoretically derived strains [12]. Inclusion of the boundary dissipation into the model predicts the behavior of the single crystal barium titanate more realistically than neglecting it. Without the boundary dissipation, the strain predicted is much higher and the polarization hysteresis is much narrower at the ends when compared to the experimentally observed values for all set of compressive stresses. For a specific case (r = 1.07 MPa), the improvement in predicting the response of thin ferroelectric single crystals by incorporating the boundary dissipation in terms of the polarization and strain vs electric field is demonstrated in Fig. 2. Under uniaxial loading, the external electric field and the compressive stress are applied parallel to one of the domain axis considered as reference axis. All the crystal variants present in the single crystal can be transformed into a single variant by application of the electric field along the reference axis. In the case of multiaxial loading, the external fields are applied at an angle to the reference axis of the crystal and this loading is referred to as the polarization rotation test. The response of the ferroelectric single crystals largely depends on the difference in the angle between

0.3

1.2

0.9

0.1

Strain (%)

Polarization (C/m2)

0.2

0 −0.1

0.3

without K with K exp

−0.2 −0.3

0.6

−10

−5

0

5

Electric field (kV/cm)

10

without K with K exp

0

−10

−5

0

5

10

Electric field (kV/cm)

Fig. 2. Effect of boundary dissipation on the hysteresis and butterfly curves of ferroelectric single crystal BaTiO3 under uniaxial electromechanical loading with constant compressive stress, r = 1.07 MPa; (solid – simulated with boundary effect; axis line – simulated without boundary effect; dashed – experiment [25]).

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K. Jayabal et al. / Computational Materials Science 44 (2009) 1222–1230

the external fields and the reference axis. Multiaxial experimental results on ferroelectric single crystals are not available in the literature and, here, an attempt is made to predict its nonlinear behavior under different loading conditions incorporating the boundary dissipation. The response of a ferroelectric single crystal under a cyclic electric field for various polarization rotations are simulated and presented in Fig. 3a and b in terms of the strain and polarization.

a

When the cyclic electric fields are applied at angles in the multiples of 90° to the reference axis of the crystal, once the stabilization is attained, the material responds in the same way as it behaves for h = 0°. For these rotations, it is evident that the crystal undergoes only 180° switching, after stabilization, from the fact that the remnant macroscopic strain remains unaltered. It was observed during the simulation of the polarization rotation that the magnitude of the polarization keeps decreasing as the angle varies from 0° to

b

0.3

1

0.1

Strain (%)

Polarization (C/m2)

0.2

1.2

0 −0.1

0.8

0.6

θ = 0 & 90°

θ = 0 & 90°

θ = 30 & 60°

−0.2

0.4

θ = 30 & 60°

θ = 45°

−0.3

−10

−5

0

5

θ = 45°

0.2

10

−10

c

0.3

d σ=

−5

0

5

10

Electric field (kV/cm)

Electric field (kV/cm)

1.78MPa

σ = 1.78 MPa

1.1

2

Polarization (C/m )

0.2 0.1 Strain (%)

0.9

0

0.7

θ = 0 & 90°

−0.1

θ = 30 & 60°

−0.2

0.5

θ = 0 & 90°

θ = 45°

−0.3

−10

−5

0

5

θ = 30 & 90° θ = 45°

0.3

10

−10

Electric field (kV/cm)

e

0

5

10

Electric field (kV/cm)

f

0

1

0

1

Stress (MPa)

Stress (MPa)

−5

θ = 0°

2

θ = 30° θ = 45°

θ = 0°

2

θ = 30° θ = 45°

θ = 60°

θ = 60°

θ = 90°

3

3

3.5 0

0.1

0.2 2

Polarization (C/m )

0.3

3.5

θ = 90° 0

0.3

0.6

0.9

1.2

Strain (%)

Fig. 3. Multiaxial response of ferroelectric single crystal BaTiO3 in terms of the hysteresis and butterfly curves under electrical (a and b), electromechanical (c and d) and ferroelastic (e and f) loading cases incorporating boundary dissipation.

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45° and it attains the lowest value at 45°. Beyond 45°, the polarization grows steadily until it reaches the maximum value at 90° similar to the polarization at h = 0°. As far as the macroscopic strain is concerned, only a linear response is realized for the angle of rotations 0° and 45°. As seen earlier, the 180° domain switching is the reason for the linear behavior of the crystal at h = 0° under cyclic electric field. But, at h = 45°, the linear response is realized by both 90° and 180° switching as both the switching process will reorient the domains at the same inclinations with respect to reference axis. However, when the angle is between 0° and 45°, owing to difference in the orientation of the domains with the reference axis, appreciable macroscopic strain is observed during the cyclic electric field. It is to be noted that the polarization and strain response of the ferroelectric single crystal when the angle of rotation is between 45° and 90° is a replica of what is observed between 0° and 45°. For instance, the ferroelectric single crystal behavior is alike at h = 30° and h = 60°. This is due to the fact that the domains are reoriented during the domain switching in such a way that their inclinations with the reference axis is the same for both the angle of rotations but in the opposite sides of the reference axis. It is to be remembered here that the boundary dissipation is not effected in the model under pure electric loading since there is no pressure on the crystal surface. The ferroelectric single crystal response under cyclic electric field in the presence of constant compressive stresses significantly varies from that of pure electric loading response as shown in Fig. 3c and d. Application of the compressive stresses produce a bulge at the corners of the polarization hysteresis which is developed by a sudden and incremental occurrence of 180° and 90° domain switching, respectively [12]. The bulge and the magnitude of the polarization get reduced with increase in the angle of rotation as shown in Fig. 3c. The magnitude of the actuation strain also decreases with increase in the angle of rotation and reaches its least value at 45° as presented in Fig. 3d. Widening of the strain hysteresis attains its peak somewhere in between 0° and 45°. The response of a ferroelectric single crystal for polarization rotations under a varying compressive stress is simulated in Fig. 3e and f. In this multiaxial ferroelastic loading, the compressive stress is applied at various angles to the reference axis. When the compressive stress is applied, the variant parallel to the reference axis gets transformed into any of the other four variants that are aligned normal to the reference axis. As this transformation takes place for h ranging from 0° to 60° in different volumes, a noticeable change in the polarization and strain occurs. After 60°, the response of the crystal turns out to be reversible and linear since

the domain switching can not take place at all. As well, when the domain switching attains a saturated state for h = 0° at a lower compressive stress for the considered K value, the saturation was not realizable for more angle of rotations even for higher value of compressive stresses. 3.3. Multiaxial response of polycrystals In ferroelectric polycrystals, single domain state can not be achieved owing to different crystallographic directions of the grains. Application of the electric field will force the domains in the grains to reorient in such a way that their spontaneous polarizations are as close to the direction of the electric field as possible. Uniaxial electromechanical experiments conducted by Lu et al. [3] on ferroelectric ceramic (PZT) was considered for recognizing the importance of the introduction of the pressure dependent boundary effects in the model developed. The model predicts that the boundary effects assume greater significance primarily when the PZT is brought under high compressive stresses and its influence may be less realized for pure electrical loading and lower value of compressive stresses. It may be reasoned as that the domains present at the grain boundaries will have to overcome a larger resistance under high compressive stresses. Hence, the grain boundary effects may considerably depend upon the externally applied stress. When the boundary effects are not included, the polarization and strain hysteresis are contracting to much lower values than the experimentally observed ones especially at high compressive stresses. Consideration of the pressure dependent boundary effects predicts more realistic behavior for ferroelectric polycrystals under electromechanical loading conditions as shown in Fig. 4. For multiaxial loading, the ferroelectric ceramics are initially poled with a strong electric field and then the external fields are applied at angles to the initial poled direction, which is considered as the reference axis. It was observed earlier that the cyclic electric fields produced significantly different polarization and strain hysteresis with respect to the angle of rotations for ferroelectric single crystals. On the contrary, the cyclic electric fields, after reaching saturation, brings out the same macroscopic response for the ferroelectric polycrystals for any angle of rotation. Hence, the polarization rotation tests are generally conducted on the poled ferroelectric polycrystals by enforcing only a part of the complete cycle [5,7]. The changes in the polarization and strain with respect to the angle of rotations were obtained as shown in Fig. 5a and b. As the angle varies from 0° to 180°, the polarization of the ferroelectric polycrystal along the direction of the electric field keeps

0.3

0

σ = −40 MPa

0.1

Strain (%)

2

Polarization (C/m )

σ = −40 MPa

0

−0.1

−0.2 −0.1

−0.3

without K with K exp

without K with K exp

−1

−0.5

0

0.5

Electric field (MV/m)

1

−0.3

−1

−0.5

0

0.5

1

Electric field (MV/m)

Fig. 4. Effect of boundary dissipation on the hysteresis and butterfly curves of ferroelectric polycrystal PZT under uniaxial electromechanical loading with constant compressive stress, r = 40 MPa; (solid – simulated with boundary effect; dotted – simulated without boundary effect; axis line – experiment [3]).

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a

b

0.6

0.6 900

180 0

0.4

135 0

0

45

Δ Strain (%)

ΔP (C/ m2)

0.4 900

45

0.2

0

0

1350

0.2

00 1800

0

0

0.2

0

c

0.5

1 1.5 Electric field (MV/m)

2

0.4

2.5

0.6

d σ = 30 MPa

0.5

1 1.5 Electric field (MV/m)

σ = 30 MPa

00 0 45 0 180

0.1 0

Δ Strain (%)

ΔP (C/ m2)

135

0

90

0.2

0 0

0

90

0

0.1

0.2 0.5

1 1.5 Electric field (MV/m)

2

2.5

0

f

0

0.5

1 1.5 Electric field (MV/m)

2

2.5

0

20

40 =0

0

= 30

60

Stress (MPa)

20

Stress (MPa)

0

135

450 00

e

2.5

0.2

1800

0.4

2

= 45

0 0

40 =0

= 30 0

60

= 45 0 = 60 0

= 60 0

80

= 90

0

= 90 0

80

0

0.05 0.1 0.15 Polarization (C/m 2)

0

0.2

0.6

0.4

0.2 0 Strain (%)

0.2 0.3

Fig. 5. Multiaxial response of ferroelectric polycrystals PZT in terms of the hysteresis and butterfly curves under electrical (a and b), electromechanical (c and d) and ferroelastic (e and f) loading cases incorporating boundary dissipation.

on growing with its maximum value attaining at h = 180°. However, this is not true with the strain change. The macroscopic strain increases steadily till the angle of rotation reaches 90° through out the electric loading. But, when the angle of rotation goes beyond 90°, the strain initially drops for some time after which it improves. This macroscopic behavior is in line with the underlying microscopic mechanism. The 90° domain switching always increases the macroscopic strain for the angle of rotations less than 90° and it may not be the case after h crosses 90° mark. Especially as h approaches 180°, the first 90° domain switching re-

duces the strain whereas the subsequent 90° domain switching increases it. The predicted response for various angles of rotations under electric fields is found to be in correlation with the experimental observations as presented in Fig. 1 [7]. However, the multiaxial behavior of ferroelectric polycrystals under electromechanical loading cases are not available in the literature and an attempt is made to understand its behavior with the proposed model. The multiaxial behavior of the PZT when an electric field is applied at various angles of rotations in the presence of a compres-

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sive stress is simulated in Fig. 5c and d. They are qualitatively similar to the plots obtained in the absence of stress, however, the influence of the stress is realized from the shift in the starting point of the polarization and the strain change, and from the reduction in their magnitudes. It is to be noted here that the variation in the polarization is more noticeable than in the strain. This may be explained as the presence of the compressive stress will provide more resistance to 90° domain switching than 180° domain switching, thus producing more changes in the polarization. The response of the ferroelectric polycrystal under polarization rotations with the compressive stresses is shown in Fig. 5e and f which has resemblance with the response of the ferroelectric single crystals. With h increasing from 0° to 90°, the size of the hysteresis for the polarization and strain decreases and a complete reversible behavior is realized at h = 90°. 4. Conclusions The dependence of the nonlinear response of the ferroelectric single and polycrystals on the orientation of the electric field and mechanical stress with regard to the initial poled direction has been studied in this paper. The model considered here is thermodynamically consistent and the pressure dependent boundary effects are included through the switching criterion. The influence of the pressure dependent boundary dissipation on a ferroelectric single crystal was correlated with the dissipation experienced by a single grain on its boundary due to the surrounding grains in a ferroelectric polycrystals under the compressive stresses. This model is able to predict the nonlinear response of ferroelectric single crystals with boundary constraints and ferroelectric polycrystals under multiaxial loading conditions while preserving the computational advantage. Sufficient experiments with different friction conditions are to be carried out to recognize the exact dependence of the behavior of thin ferroelectric single crystals on the boundary effects for validation and further improvements in

the model. Similarly, in the case of the polycrystals, multiaxial loading with different grain sizes may provide the required details to understand the range of applicability of the model considered in this work. References [1] S.C. Hwang, C.S. Lynch, R.M. McMeeking, Acta Metall. Mater. 43 (1995) 2073– 2084. [2] C.S. Lynch, Acta Mater. 44 (1996) 4137–4148. [3] W. Lu, D.N. Fang, C.Q. Li, K.C. Hwang, Acta Mater. 47 (1999) 2913–2926. [4] J. Fan, W.A. Stoll, C.S. Lynch, Acta Mater. 47 (1999) 4415–4425. [5] J.E. Huber, N.A. Fleck, J. Mech. Phys. Solids 49 (2001) 785–811. [6] J. Shieh, J.E. Huber, N.A. Fleck, Acta Mater. 51 (2003) 6123–6137. [7] D. Zhou, M. Kamlah, B. Laskewitz, Proc. SPIE. 6170 (2006) 617009. [8] W. Chen, C.S. Lynch, J. Eng. Mater. Technol. 123 (2001) 169–175. [9] S.E. Park, T.R. Shrout, J. Appl. Phys. 82 (1997) 1804–1811. [10] T. Liu, C.S. Lynch, J. Appl. Phys. 51 (2003) 407–416. [11] K. Bhattacharya, G. Ravichandran, Acta Mater. 51 (2003) 5941–5960. [12] D. Shilo, E. Burcsu, G. Ravichandran, K. Bhattacharya, Int. J. Solids Struct. 44 (2007) 2053–2065. [13] M. Kamlah, Continuum Mech. Thermodyn. 13 (2001) 219–268. [14] C.M. Landis, Curr. Opin. Solid State Mater. Sci. 8 (2004) 59–69. [15] J.E. Huber, Curr. Opin. Solid State Mater. Sci. 9 (2005) 100–106. [16] T. Michelitsch, W.S. Kreher, Acta Mater. 46 (1998) 5085–5094. [17] W. Chen, C.S. Lynch, Acta Mater. 46 (1998) 5303–5311. [18] S.C. Hwang, J.E. Huber, R.M. McMeeking, N.A. Fleck, J. Appl. Phys. 84 (1998) 1530–1540. [19] J.E. Huber, N.A. Fleck, C.M. Landis, R.M. McMeeking, J. Mech. Phys. Solids 47 (1999) 1663–1697. [20] A. Arockiarajan, B. Delibas, A. Menzel, W. Seemann, Comput. Mater. Sci. 37 (2006) 306–317. [21] K. Jayabal, A. Arockiarajan, S.M. Sivakumar, Comput. Model. Eng. Sci. 27 (2008) 111–123. [22] M. Kamlah, Q. Jiang, Smart Mater. Struct. 8 (1999) 441–459. [23] M. Selten, G.A. Schneider, V. Knoblauch, R.M. McMeeking, Int. J. Solids Struct. 42 (2005) 3953–3966. [24] C.T. Sun, A. Achuthan, Proc. SPIE. 4333 (2001) 240–249. [25] E. Burcsu, G. Ravichandran, K. Bhattacharya, J. Mech. Phys. Solids 52 (2004) 823–846. [26] M.G. Shaikh, S. Phanish, S.M. Sivakumar, Comput. Mater. Sci. 37 (2006) 178– 186.