Finite Element Modeling in Analyzing Physical Properties of the Pb-Free Piezoelectric Materials

Abstract

New giant piezoelectric factor materials such as PZT, PZN-PZT, PMN-PT and PZN-PT etc. were researched during the last decade and are actually becoming commercially available. However, there is an urgent demand for high performance Pb-free piezoelectric to substitute for the current workhorse, the PZT family. Recently, many Pb-free systems have been reported that shows equally as excellent piezoelectricity as materials in PZT family at room temperature. In this present study, it uses the Finite Element modeling (FEM method) with the simulation program to compare the physic properties of piezoceramics in PZT family and Pb-free piezoceramic Ba(Zr 0.2 Ti 0.8)O 3-50(Ba 0.7 Ca 0.3)TiO 3 (BZT-50BCT) and show the advantage properties of BZT-50BCT. Furthermore, the results will be a premise of using the simulation method in researching the properties of new piezoelectric materials to shorten time as well as save money and time.

Full text

Journal of Materials Science and Engineering A 3 (4) (2013) 283-289
Finite Element Modeling in Analyzing Physical
Properties of the Pb-Free Piezoelectric Materials
Vo Thanh Tung1, Nguyen Trong Tinh2, Nguyen Hoang Yen1, Le Thi Ngoc Bao1 and Dang Anh Tuan1
1. Hue University of Science, Hue University, Hue City 08454, Vietnam
2. Institute of Applied Physics and Scientific, Instrument of Vietnamese Academy of Science and Technology, 08404, Vietnam
Received: November 27, 2012 / Accepted: December 18, 2012 / Published: April 10, 2013.
Abstract: New giant piezoelectric factor materials such as PZT, PZN-PZT, PMN-PT and PZN-PT etc. were researched during the
last decade and are actually becoming commercially available. However, there is an urgent demand for high performance Pb-free
piezoelectric to substitute for the current workhorse, the PZT family. Recently, many Pb-free systems have been reported that shows
equally as excellent piezoelectricity as materials in PZT family at room temperature. In this present study, it uses the Finite Element
modeling (FEM method) with the simulation program to compare the physic properties of piezoceramics in PZT family and Pb-free
piezoceramic Ba(Zr0.2Ti0.8)O3-50(Ba0.7Ca0.3)TiO3 (BZT-50BCT) and show the advantage properties of BZT-50BCT. Furthermore, the
results will be a premise of using the simulation method in researching the properties of new piezoelectric materials to shorten time
as well as save money and time.
Key words: BZT-50BCT, finite element modeling, PZT family.
1. Introduction
Piezoelectric materials are an important class of
functional materials that create an electrical
voltage/charge when mechanically stressed/strained [1,
2]. For half a century, the PZT (lead zirconate titanate)
family has been the icon of a large class of
technologically important materials piezoelectrics,
which dominated almost all piezoelectric applications
ranging from cell phone to high-tech
scanning-tunneling microscope [3, 4]. Unfortunately,
PZT is now facing global restrictions in its usage
because of Pb toxicity to the environment and to
human body. Therefore there is an urgent need to
develop Pb-free piezoelectric materials that can
compete with PZT family.
Recently many Pb-free ferroelectric systems, such as
Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 (hereafter abbreviated
Corresponding authors: Vo Thanh Tung, Ph.D., research
field:s atomic forcr microscope (methods and applications),
simulation and developtment of instruments. Email:
votungbeo@gmail.com.
as BZT-xBCT), (K0.436Na0.5Li0.064)Nb0.92Sb0.08O3 etc.
have been reported. However, the BZT-xBCT material
is particularly interested because it exhibits the
equally excellent piezoelectricity as in Pb-based
systems [5].
In the present study, we used a full set of elastic,
dielectric, and piezoelectric constants (tensor
components) for the BZT-50BCT ceramic, pure
BatiO3, the soft PZT ceramic (PZT5A) and the
multi-component ceramic PZN-PZT which defined by
theoretically and experimentally. We use the finite
element modeling with the simulations program to
analyze, compare the physic properties of
piezoceramics in PZT family and Pb-free
piezoceramic Ba(Zr0.2Ti0.8)O3-50(Ba0.7Ca0.3)TiO3
(BZT-50BCT) and show the advantage properties of
BZT-50BCT. Greater knowledge in BZT-50BCT
ceramic allow for improved Pb-free piezoelectric
material with the interest specific applications. Further,
from the measured results, we thus suggest the
combination of the using the simulation method in
DAVID PUBLISHING
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Finite Element Modeling in Analyzing Physical Properties of the Pb-free Piezoelectric Materials
284
researching the properties of new piezoelectric
materials to shorten time as well as money.
2. Finite Element Modeling Theory
The mechanical properties of piezoelectric
materials (the relation between stress and strain) can,
assuming only small deformation, be described as a
linear-elastic material. It assumes that the strains are
small and the material follows Hooke’s law (i.e. is
elastic). This relation for an arbitrary material will
then be expressed by the generalized Hooke’s law
which is shown by the matrix form.
For mathematical description of piezoelectric
materials, these are one of four sets of piezoelectric
coefficients used to relate the mechanical (S and T)
and the electrical variables (E and D). A permittivity
matrix, ε, has also been introduced to replace the
vacuum permittivity and the polarization vector. And
it has been necessary to include the coefficients. All
these relations are listed in their abbreviated form in
the below equations giving four different sets of
piezoelectric parameters.
D = eS + εSE; T = cES  etE
D = dT + εTE; S = sET + dtE
E = -gT + (εT)-1D; S = sDT + gtD
E = -hS + (εS)-1D; T = cDS  htD
Each of these sets of parameters gives a
mathematical description of the direct and the
inverse piezoelectric effect in the upper and lower
equation respectively, and is sufficient to describe
the material. Each of the four sets are also related to
each other, hence it is possible to change from one
set of parameters to another if it should become
necessary.
These equations contain a large number of
independent variables. However, this number is
significantly reduced when including the symmetry in
piezoelectric materials, thus giving the following 3-D
material matrices.
Elastic compliance for constant electric field, Sij
E
and Elastic stiffness constants, Cij:
EEE
11 12 13
EEE
12 11 13
EEE
13 13 33
E
44
E
44
E
66
S S S 0 0 0
S S S 0 0 0
S S S 0 0 0
0 0 0 S 0 0
0 0 0 0 S 0
0 0 0 0 0 S




















EEE
11 12 13
EEE
12 11 13
EEE
13 13 33
E
44
E
44
66
C C C 0 0 0
C C C 0 0 0
C C C 0 0 0
0 0 0 C 0 0
0 0 0 0 C 0
0 0 0 0 0 CE











Piezoelectric coefficients dij and eij:
15
15
31 31 33
0 0 0 0 d 0
0 0 0 d 0 0
d d d 0 0 0










15
15
31 31 33
0 0 0 0 e 0
0 0 0 e 0 0
e e e 0 0 0





From the theory, the material properties are defined
in the stress-charge form, in which the user has to
specify the elasticity matrix, the coupling matrix, the
relative permittivity matrix, the piezoelectric
coefficients matrix and the density. These matrices of
these parameters are defined as in Table 1. The
complete set of constants of the pure BaTiO3 ceramic,
the PZT5A material shown in Table 1 is taken out
from the data of the libraries of the simulation
program. A full set of elastic, piezoelectric, and
dielectric parameters of BZT-50BCT are measured
and calculated by Xue, et al. [6] when using a
resonance method. About the PZN-PZT, we have
calculated from the data of some the
experimental results [7] and using all the formulas in
the IEEE standard for piezoelectric ceramics
characterization [8] that have been exclusively used in
the past.
Investigations using computational finite element
methods (FEM method) have successfully analyzed
the behavior of piezoceramic discs [9-10]. In our
study, we used a simplified 3-D axis-symmetric
model in the FEM of simulation program for design
the model geometry. The modeling dimensions of
piezoceramic discs with radius 5.5 mm, thickness 0.6
mm is illustrated in Fig. 1a. Modeling conducted is
meshed of by using the standard meshing tool (the
free tetrahedral mode) at 13,380 elements, 16
numbers of vertex elements, 208 numbers of edge

Finite Element Modeling in Analyzing Physical Properties of the Pb-free Piezoelectric Materials
285
(a) (b)
Fig. 1 (a) The fully expanded model of the disk ceramic samples, (b) the meshed model of the designed disk ceramic samples.
Table 1 Measured and derived piezoelectric, dielectric constants and elastic constants of poled BZT-50BCT ceramic
(density 5,200 kg/m3) compared to the BaTiO3 (density 5,720 kg/m3) ceramic, the PZT-5A ceramic (density 7,750 kg/m3) and
PZN-PTZ ceramic (density 7,720 kg/m3).
Elastic Stiffness Constants, cij (1010 N/m2)
Sample C11
E C12
E C
13
E C33
E C44
E C66
E
BZT-50BCT 13.6 8.9 8.5 11.3 2.66 2.44
BaTiO3 15.037 6.56308 6.59391 14.5521 4.38596 4.23729
PZT5A 12.1 7.7 7.7 11.1 2.1 2.3
PZN-PZT 12.47 12.43 13.64 15.01 4.89 2.09
Elastic Compliance Constants, sij (10-12 m2/N)
Sample S11
E S12
E S
13
E S33
E S44
E S66
E
BZT-50BCT 15.5 -5.5 -7.4 19.7 37.6 42.0
BaTiO3 9.1 -2.7 -2.9 9.5 22.8 23.6
PZT5A 16.4 -5.74 -7.22 18.8 47.5 44.3
PZN-PZT 17.15 -6.71 -9.49 17.33 20.42 47.71
Piezoelectric Coefficients
d
ij (10-12 N/C) eij (C/m2)
Sample d33 d
31 d
15 e33 e31 e15
BZT-50BCT 546 -231 453 22.4 -5.7 12.1
BaTiO3 190 -78 260 17.3624 -4.32015 11.4035
PZT5A 374 -171 584 15.8 -5.4 12.3
PZN-PZT 499 -244 322 8.2 -7.22 15.7
Dielectric constants, εij (ε0)
Sample ε33
T ε11
T ε33
S ε11
S
BZT-50BCT 4,050 2,732 2,930 1,652
BaTiO3 1,700 1,450 1,251.3 1,115.1
PZT5A 1,700 1,730 830 916
PZN-PZT 3,843 2,715 1,079 782
elements and 4,352 numbers of boundary elements in
Fig. 1b.
Since the analysis involves piezoelectric material,
there are two different set of boundary conditions the
user has to set. First is the mechanical boundary
conditions, second is the electrical boundary
conditions. In terms of the mechanical boundary
conditions, all surfaces are free. For the electrical
boundary condition, the bottom surface of the
cantilever beam is set as ground, the top surface set as
terminal/electrical with the voltage 0.5 V. These
boundary conditions are illustrated in Fig. 1a.
The dependence of total displacement on the
frequency change of the PZT family and BZT-50BCT

Finite Element Modeling in Analyzing Physical Properties of the Pb-free Piezoelectric Materials
286
(a) (b)
Fig. 2 Comparisons of FEM model of the frequency dependence of the total displacement of the ceramic samples for
materials in PZT family and Pb-free family (a) 10 KHz to 3.2 MHz and (b) 10KHz to 1 MHz.
(a) (b)
(c) (d)
Fig. 3 Modal shapes generated from computational analysis of the ceramic samples for materials in PZT family and Pb-free
family, as related to analysis in Fig. 2 in low frequencies, the radial vibration modes: (a) 0.3PZN-0.7PZT and (b) BZT-50BCT;
(c) BaTiO3 and (d) PZT-5A.

Finite Element Modeling in Analyzing Physical Properties of the Pb-free Piezoelectric Materials
287
(a) (b)
Fig. 4 Vibration modes from computational analysis of the ceramic samples for materials BaTiO3 and Pb-free family, as
related to analysis in Fig. 2 in high frequencies, the thickness vibration modes. (a) BaTiO3 and (b) BZT-50BCT.
were investigated in the FEM method. Further, the
results of the analysis such as the piezoelectric
parameters of piezoceramics, vibration modes,
electrical potential and other parameters of all these
materials were performed.
4. Simulation Results and Discussion
Fig. 2 shows a comparison the dependence of the
total displacement on frequency for piezoceramics of
the PZT family and BZT-50BCT materials
respectively. It is firstly seen that the total
displacement for BZT-50BCT ceramics are rather low
(about 0.9 μm at high resonance frequency, 2.5 MHz),
however, it is higher when comparing to ceramics of
the BaTiO3 and PZT5A materials (Fig. 2). The data in
Fig. 2 shows that the displacement at high resonance
frequency of piezoceramics BZT-50BCT is almost
three times as large as that of the BaTiO3 and PZT5A
ceramics. Further, the appearance of peaks of the total
displacement of the BZT-50BCT ceramics is observed
for the other resonance frequencies, same as the other
piezomaterials (an example at the low frequencies in
Fig. 2b). The simulation results in the total
displacement research of the BZT-50BCT ceramics
when have the applied voltage is an important
evidence that it has high piezoelectric properties.
Using the scripting capabilities in simulation
program, the modal shapes for the ceramics at the low
resonant frequencies highlighted in Fig. 2b can be
visualized in Fig. 3 which illustrates the sum of
displacement generated during vibration. From the
results in the modal shapes in Fig. 3, we can know
exactly the relation between the total displacements at
the different positions in the piezoceramics disk as
well as to perform a meaningful comparison of the
physic properties between these materials.
The results in Fig. 3 display the radial vibration
modes of the ceramic samples in the low frequencies
that allow for the relation between fundamental
resonance frequency and its dimensional identity and
harmonics. Analyzing of modal shapes from Fig. 3a to
Fig. 3d, it is seen that the total displacement of
PZN-PZT materials is very high, about 3  10-3
mm at
the resonant position, but the surface deformation is
not smooth (Fig. 3c). The rough appears in the surface
of ceramic disk PZN-PZT. On the contrary, the
surface deformation of three other specimens is very
good, especially; the total displacement of these
specimens is equivalent at the low resonant
frequencies.
The vibration modal shapes for the ceramic samples
at the high resonant frequencies highlighted in Fig. 2a
can be visualized in Fig. 4. They operate in the
thickness vibration modes with the resonant frequency

Finite Element Modeling in Analyzing Physical Properties of the Pb-free Piezoelectric Materials
288
(a) (b)
Fig. 5 The dependence of the impedance Z on frequencies of the ceramic samples for materials in PZT family and Pb-free
family (a) 10 KHz to 3.2 MHz (b) 10 KHz to 1 MHz using FEM method.
about 2.91 MHz for piezoceramic PZT-50BCT sample
and about 2.36 MHz for BaTiO3 sample. As seen,
more than one harmonic vibration modes appears
beside the fundamental resonance frequency.
Furthermore, the results show that maximum total
displacement of BZT-50BCT specimen is similar to
the specimens of the materials in PZT family, example
BaTiO3 sample.
The results are in qualitative agreement and they
indicate that the piezoelectric properties of Pb-free
piezoceramic PZT-50BCT are equivalent to the
piezoceramics in PZT family, maybe it’s better.
Secondly, the frequency dependences of impedance Z
from 10 KHz to 3.5 MHz and from 10 KHz to 1 MHz
of these ceramic samples are shown in Figs. 5a and 5b.
It is found that the calculated results agree well with
the experimental results for the materials in PZT
family. From the received results, it shows that the
appearance of the resonant peaks in the simulation
curve and minimum-to-maximum impedance ratio of
the BZT-50BCT specimen are similar to the
specimens of the materials in PZT family. These
analyses prove the important role in the high
piezoelectricity of Pb-free piezoceramic PZT-50BCT.
4. Conclusions
In summary, we developed of the FEM using
simulation program to perform a Pb-free piezoelectric
system BZT-50BCT, which exhibits high
piezoelectric properties, which is comparable even
with the materials in PZT family. The full results in
simulation were obtained that are of importance for
both practical device design and fundamental study. In
addition, it is found that the FEM model connected the
simulation program has excellent agreement to the
experimental data of the researched material
groupings. It will highlight the trend of combination
the FEM model and experimental methods in studying
the properties of the new material groups.
Acknowledgments
The work was carried out in the frame of the “Basic
Project of Hue University 2013-2015”.
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