Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations
Biomedical transducers widely employ piezoceramic polymer composites with 1-3 connectivity. The research is aimed at determining the effective material properties of these composites in the micrometer scale by simulation. Volume fraction of piezoceramic in the composite plays an important role in co...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2010
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/32047 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations / C.V. Madhusudhana Rao, G. Prasad // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13703: 1-10. — Бібліогр.: 21 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-32047 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-320472012-04-07T12:22:21Z Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations Madhusudhana Rao, C.V. Prasad, G. Biomedical transducers widely employ piezoceramic polymer composites with 1-3 connectivity. The research is aimed at determining the effective material properties of these composites in the micrometer scale by simulation. Volume fraction of piezoceramic in the composite plays an important role in composite material properties as studied by several researchers. It is also noted that the fiber aspect ratio (a/l) also affects the composite material properties. Therefore it is intended to determine the effective material properties both analytically and by simulation using computer simulation software program ANSYS which implements finite element method (FEM). In the present work piezoelectric fiber composites are modeled, analyzed and then the results are verified by using analytical equations. Important conclusions are drawn to select the piezocomposite materials for transducer applications. 2010 Article Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations / C.V. Madhusudhana Rao, G. Prasad // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13703: 1-10. — Бібліогр.: 21 назв. — англ. 1607-324X http://dspace.nbuv.gov.ua/handle/123456789/32047 PACS: 77.65.-j, 77.84.-s, 77.65.-j, 43.38.Fx, 47.11.Fg en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Biomedical transducers widely employ piezoceramic polymer composites with 1-3 connectivity. The research is aimed at determining the effective material properties of these composites in the micrometer scale by simulation. Volume fraction of piezoceramic in the composite plays an important role in composite material properties as studied by several researchers. It is also noted that the fiber aspect ratio (a/l) also affects the composite material properties. Therefore it is intended to determine the effective material properties both analytically and by simulation using computer simulation software program ANSYS which implements finite element method (FEM). In the present work piezoelectric fiber composites are modeled, analyzed and then the results are verified by using analytical equations. Important conclusions are drawn to select the piezocomposite materials for transducer applications. |
| format |
Article |
| author |
Madhusudhana Rao, C.V. Prasad, G. |
| spellingShingle |
Madhusudhana Rao, C.V. Prasad, G. Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations Condensed Matter Physics |
| author_facet |
Madhusudhana Rao, C.V. Prasad, G. |
| author_sort |
Madhusudhana Rao, C.V. |
| title |
Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations |
| title_short |
Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations |
| title_full |
Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations |
| title_fullStr |
Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations |
| title_full_unstemmed |
Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations |
| title_sort |
characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/32047 |
| citation_txt |
Characterization of 1-3 piezoelectric polymer composites -- a numerical and analytical evaluation procedure for thickness mode vibrations / C.V. Madhusudhana Rao, G. Prasad // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13703: 1-10. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT madhusudhanaraocv characterizationof13piezoelectricpolymercompositesanumericalandanalyticalevaluationprocedureforthicknessmodevibrations AT prasadg characterizationof13piezoelectricpolymercompositesanumericalandanalyticalevaluationprocedureforthicknessmodevibrations |
| first_indexed |
2025-07-03T12:31:52Z |
| last_indexed |
2025-07-03T12:31:52Z |
| _version_ |
1836629006682685440 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13703: 1–10
http://www.icmp.lviv.ua/journal
Characterization of 1-3 piezoelectric polymer
composites – a numerical and analytical evaluation
procedure for thickness mode vibrations
C.V. Madhusudhana Rao∗, G.Prasad†
Department of Physics, Osmania University, Hyderabad – 500 004 A.P, India
Received August 13, 2009, in final form February 23, 2010
Biomedical transducers widely employ piezoceramic polymer composites with 1-3 connectivity. The research
is aimed at determining the effective material properties of these composites in the micrometer scale by
simulation. Volume fraction of piezoceramic in the composite plays an important role in composite material
properties as studied by several researchers. It is also noted that the fiber aspect ratio (a/l) also affects the
composite material properties. Therefore it is intended to determine the effective material properties both
analytically and by simulation using computer simulation software program ANSYS which implements finite
element method (FEM). In the present work piezoelectric fiber composites are modeled, analyzed and then
the results are verified by using analytical equations. Important conclusions are drawn to select the piezocom-
posite materials for transducer applications.
Key words: piezoelecticity, piezoelectic materials, electromechanical effects, Piezoelectric transducers,
Finite element methods
PACS: 77.65.-j, 77.84.-s, 77.65.-j, 43.38.Fx, 47.11.Fg
1. Introduction
Piezoelectric composites have attracted the attention of many researchers for the past several
decades due to their striking features in practical applications in ultrasonic transducers. Piezoelec-
tric transducers convert electrical signals into mechanical signals and vice versa. Piezocomposites
are widely used in many fields such as ultrasonic transducers, biomedical sensors, MEMS such as
water purity detection sensors etc. Ferroelectric ceramics have been extensively employed in a verity
of such sensors and actuators [1]. They serve as transmitters and receivers in the imaging systems
for sonar, medical, and NDE (nondestructive evaluation) applications, as well as in non-imaging ap-
plications like SAW (surface acoustic wave) devices in signal processing. In such applications, bulk
ferroelectric ceramic materials have certain shortcomings. Piezoceramics have very high electrome-
chanical coupling coefficient (k). However, they have very high acoustic impedance and they are
very brittle which affects fabrication. For this reason fiber piezocomposites are developed by combi-
nation of piezoceramics and polymers from the piezocomposites. These composites have shown re-
markable improvement in the performance of ultrasonic devices over their counterparts like piezoce-
ramics and piezopolymers with their tailor-made properties. They have suitable material properties
for transducer applications such as moderate electromechanical coefficients, low acoustic impedance
and low density, wide range of dielectric constants, low dielectric loss and low mechanical loss [2].
Variety of piezocomposite materials can be made by combining a piezoceramic with a passive
polymer such as epoxy or active polymer such as PVDF [3]. The piezocomposites are classified
according to their connectivity (such as 2–2, 1-3, 0–3 etc.,). Connectivity is defined as the number
of dimensions through which the material is continuous [4]. It is conventional for the first digit to
refer to the piezoelectrically active phase. These achievements are made possible with enormous
research in the design and investigation of the piezocomposite materials. Experimental work done
∗E-mail: cvm rao@rediffmail.com
†E-mail: gudurup@osmania.ac.in
c© C.V. Madhusudhana Rao, G.Prasad 13703-1
http://www.icmp.lviv.ua/journal
C.V. Madhusudhana Rao, G.Prasad
in the fabrication of piezocomposites, with 2–2 and 1-3 connectivity, lead to ultra-fine technologies
in their manufacturing process [5–8]. Presently, piezocomposites in the form of layers of a few
micrometers thick and fibers of a few a micrometers diameter are fabricated with the advancement
in the manufacturing technologies. The field of MEMS (Micro-Electro-Mechanical Systems) has
grown in leaps and bounds with this advent. The study of micromechanics of piezocomposites,
with their effective material properties in focus, resulted in more careful choice of materials for
specific purposes [9]. Optimization of the device performance is achieved by both simulation and
experimental evaluation methods [10]. The most commonly used method for designing composite
transducer devices is the effective medium model proposed by Smith [11–13]. The effective medium
theory predicts the material properties of the composites based on the volume fraction of the
piezoceramic enclosed in the soft matrix medium. However, the aspect ratio of the fibers (aspect
ratio is defined as a/l, where a is the diameter and l is the length of fibers) is not included in the
analytical equations derived by Smith et.al. For this reason FEM is adopted to study the effect of
aspect ratio on the material properties of composite which can be used in transducer design.
A brief description of piezocomposites is presented in the first section. The constitutive equati-
ons related to piezoelectric materials are discussed in the second section. Details of finite element
modeling are then presented in the third section. In the present study an attempt is made to
determine the material coefficients which are important in biomedical transducers. A step by step
approach is presented and discussed in the fourth section. The important conclusions drawn are
presented in the last section.
2. Piezoelectric 1-3 composites
Composite materials have applications in transducers for pulse echo medical ultrasonic imag-
ing. There are several requirements for piezoelectric used in these transducers. First, for sensitive
transducers, the piezoelectric should convert between electric and mechanical energy. Second, the
piezoelectric should be acoustically matched to tissue so that acoustic waves couple well in the
transducer and tissue both during transmission and reception. Third, the electric properties should
be compatible with the driving and receiving electronics. The requirement of relevant parameters on
high electromechanical coupling coefficient (kt → 1), acoustic impedance close to tissue (Z → 1.5
Mrayls), large dielectric constant (> 100), low dielectric loss (tan δ < 0.05) and low mechanical
loss (Qm > 10) [13].
Polymer piezoelectric composites with 1-3 connectivity can meet many of these stringent re-
quirements. In this paper we report a detailed finite element analysis using a unit cell model for
thickness mode vibrations. The results obtained are compared with the analytical and experimental
results. The agreement between the results better proves the validity of the method of simulation.
Figure 1. Rectangular piezoceramic fibers in
square arrangement [14].
Figure 2. Cylindrical piezoceramic fibers in
hexagonal arrangement [7].
Piezoelectric 1-3 composites are formed by embedding piezoelectric ceramic fibers in a passive
polymer medium. The ceramic fibers may have a square or circular cross section. They may be
arranged in square or hexagonal arrangement as shown in figures 1 and 2. The rod composite ge-
ometry provides materials with enhanced electromechanical coupling and with acoustic impedance
13703-2
Characterization of 1-3 piezoelectric polymer composites
close to that of biological tissue; these advantages yield transducers for medical ultrasonic imag-
ing with high sensitivity and compact impulse response. Proper design of the rod spacing (lateral
fine scale spacing) yields materials which exhibit low cross-talk between array elements formed by
patterning the electrode alone, without cutting between the elements. Keeping this in view, fiber
composites having very fine fiber diameter (∼ 40 micrometer) are analyzed.
3. Piezoelectric constitutive equations
Piezoelectricity is a coupling between a material’s mechanical and electrical behavior, which
can be represented by the following constitutive equations in tensor notation. Tij Stress, Skl strain,
electric field Ek and electric displacement Di are related by equation (1)
Tij = cE
ijklSkl − et
kijEk ,
Di = ekijSkl + εS
jkEk , (1)
where cE
ijkl is a fourth-order elasticity tensor under short circuit boundary conditions. εS
ik is a
second-order free body dielectric tensor and ekij is a third-order piezoelectric strain tensor. Due
to symmetry of the tensors Tij , Skl, cE
ijkl and εS
ik, the above equation (1) can be written in the
matrix notation as equation (2) as follows.
[
T
D
]
=
[
CE
−et
e εS
][
S
E
]
, (2)
where t denotes a transposed matrix.
For a transversely isotropic piezoelectric material, the stiffness matrix, the piezoelectric matrix
and the dielectric matrix simplify so that 7 of the orthotropic constants depend on the others as
given below and only 11 independent coefficients are to be determined.
c22 = c11 , c23 = c13 , c55 = c44 , c66 = (c11 − c12)/2 ,
e24 = e15 , e32 = e31 , ε22 = ε11 . (3)
Therefore, the constitutive equation (1) can be written as follows.
T̄11
T̄22
T̄33
T̄23
T̄31
T̄12
D̄1
D̄2
D̄3
=
Ceff
11 Ceff
12 Ceff
13 0 0 0 0 0 −eeff
13
Ceff
12 Ceff
11 Ceff
13 0 0 0 0 0 −eeff
13
Ceff
13 Ceff
13 Ceff
33 0 0 0 0 0 −eeff
33
0 0 0 Ceff
44 0 0 0 −eeff
15 0
0 0 0 0 Ceff
44 0 −eeff
15 0 0
0 0 0 0 0 Ceff
66 0 0 0
0 0 0 0 eeff
15 0 εeff
11 0 0
0 0 0 eeff
15 0 0 0 εeff
11 0
eeff
13 eeff
13 eeff
33 0 0 0 0 0 εeff
33
S̄11
S̄22
S̄33
S̄23
S̄31
S̄12
Ē1
Ē2
Ē3
.
(4)
A bar on T , S, E and D indicates average values and the piezoelectric coefficients are the effective
properties of the composite structure as indicated by the superscript. These equations are further
used in numerical evaluation process.
4. Analytical equations
The effective properties of the piezocomposite which are more relevant in thickness mode vi-
brations are given by below equations (6)–(17) where all the parameters are represented by IEEE
standard notations [13]. The volume fraction of the inclusion piezoceramic fiber material in the
13703-3
C.V. Madhusudhana Rao, G.Prasad
composite is represented by ν and the volume fraction of the matrix material is represented by ν,
where
ν = (1 − ν). (5)
The mean elastic constants are given by:
cE
13 =
ν cE
13(c11 + c12) + ν c12(c
E
11 + cE
12)
ν(c11 + c12) + ν(cE
11 + cE
12)
, (6)
cE
33 = ν
[
cE
33 −
2ν(cE
13 − c12)
2
ν(c11 + c12) + ν(cE
11 + cE
12)
]
+ ν c11 . (7)
The average electromechanical coefficients are given by:
e31 =
ν e31(c11 + c12)
ν(c11 + c12) + ν(cE
11 + cE
12)
, (8)
e33 = ν
[
e33 −
2ν e31(c
E
13 − c12)
ν(c11 + c12) + ν cE
11 + cE
12
]
. (9)
The average static dielectric constant is:
εS
33 = ν
[
εS
33 +
2ν(e31)
2
ν(c11 + c12) + ν(cE
11 + cE
12)
]
+ ν ε11 . (10)
From the above, one can calculate:
cD
33 = cE
33 + (e33)
2/εS
33 . (11)
Piezoelectric h constant is given by:
h33 =
e33
εS
33
. (12)
The dielectric susceptibility is inverse of permittivity and given by:
β
S
33 =
1
εS
33
. (13)
The average density of composite depends on densities of polymer (ρp) and density of ceramic (ρc)
by:
ρ = νρc + νρp . (14)
The average electromechanical coupling coefficient is given by:
kt = h33/(cD
33β
S
33)
1/2 = e33/(cD
33ε
S
33)
1/2 . (15)
Acoustic impedance is given by
z = (cD
33ρ)1/2 . (16)
Longitudinal acoustic velocity is given by:
vl = (cD
33/ρ)1/2 . (17)
Using the above equations and the values of component phase parameters, average values of com-
posite parameters are calculated and plotted.
13703-4
Characterization of 1-3 piezoelectric polymer composites
5. Numerical evaluation – implementation of finite element method
A full model of composite is difficult to analyze because of the constraints such as limited
processing speed and memory available for computation of computing devices. Therefore, for sim-
plicity, a periodic array of cylindrical fibers with square arrangement is considered as shown in
figure 3. This periodic nature of fibers in the composite allows us to select a simple volume element
which represents the general characteristics of the whole composite. This volume element is also
called a unit cell as shown in figure 4. It is this volume element that we consider for analysis
by applying periodic boundary conditions. The complete description of the evaluation of effective
material properties by simulation can be found in [15].
The boundary conditions that are applied to the unit cell to evaluate various effective material
properties are briefly described as follows.
• To apply strain in X3− direction, predefined displacements are applied perpendicularly on
C+ plane on all nodes. At the same time, zero displacements are applied on C− plane.
• To apply pure shear stress in X1X2-plane, tangential forces are applied on all nodes on A+
and B+ planes while constraining the displacement of nodes along X3-direction.
• To apply electric field along X3-direction, C− plane is kept at zero potential while a non-zero
potential is applied on C+ plane.
Figure 3. Unit cells for square arrangement of
fibers.
Figure 4. Notations for identifying surfaces of
the unit cell.
The effective properties of the piezocomposite can be extracted by solving the matrix equations
given by equation (4). We solve these equations simultaneously by applying appropriate boundary
conditions so that only one equation is set to give a non-zero result while all others vanish. For
example to evaluate eeff
33 , the following boundary conditions are applied.
• Apply zero normal displacements to all surfaces so that all the strain terms become zero.
• Apply zero potential at C− plane and a non-zero potential on C+ plane so that an electric
field exists along Z-direction.
• Apply zero potential at all other planes so that no electric field exists along X and Y-
directions.
After the above boundary conditions are applied, eeff
33 is calculated as eeff
33 = −T33/E33. All
the other effective properties are extracted by applying suitable boundary conditions.
Following Smith’s analytical model as given in [11], Chan et. al. [12] derived the analytical
equations for the effective material properties of 1-3 piezocomposites. They also experimentally
checked the validity of their equations for various physical quantities for different volume fracti-
ons of piezoceramic in the composite material. The coincidence between the theoretical and the
reported experimental values is quite appreciable. Analytical, numerical and experimental values
are compared for PZT 7A – Araldite D composites. Model’s predictions and their implications for
transducer applications are considered.
As a part of the analysis, the unit cell of 1-3 piezocomposite is modeled with the fiber mate-
rial made of PZT 7A which is included in an Araldite D matrix. The effective properties of the
13703-5
C.V. Madhusudhana Rao, G.Prasad
piezocomposite which are more relevant in thickness mode vibrations are extracted by simulating
the unit cell of the 1-3 composite and performing the static analysis by ANSYS software and they
are compared with the analytical and experimental results following Smith’s equations [13]. The
material properties of PZT 7A and Araldite D are taken from [12]. To check the correctness of the
model, the effective material properties of the composite, namely, c33, e33 and ε33 are determined
for different volume fractions of fiber material in the composite by simulation.
Figure 5. Variation of Acoustic Impedance with volume fraction.
Figure 6. Variation of kt with volume fraction.
It is found that the effective properties c33, e33 and ε33 linearly vary with volume fraction. From
these values the effective transducer parameters, namely, the electromechanical coupling coefficient,
kt, the acoustic impedance Z and the longitudinal velocity Vl are determined using equations given
in section 3. These simulation results are shown graphically in figures 5–9 along with the analytical
results following Smith’s model. Experimental values are provided for comparison wherever they
are available. The experimental values are taken from [12]. Evidently the simulation results are in
close agreement with the analytical and experimental values. In the graphs, the results obtained by
simulation are indicated by the label FEM and the values that are calculated by using the analytical
equation presented in section 3 are indicated by the label ANALYTICAL. The experimental values
are indicated by the label EXPT.
In figure 5 it can be noted that the acoustic impedance increases linearly with the increase of
volume fraction. At lower volume fractions, the variation is almost linear but at higher volume
13703-6
Characterization of 1-3 piezoelectric polymer composites
fractions the variation shows non-linear behavior as we observe the numerical results while the
analytical solution predicts completely linear behavior.
In figure 6 it can be noted that the thickness mode electromechanical coupling coefficient, kt
increases with the volume fraction because the electrical energy is more efficiently converted into
mechanical energy. There is a close agreement between the simulated and analytical results at all
volume fractions.
Figure 7. Variation of Relative permittivity with volume fraction.
Figure 8. Variation of Longitudinal velocity with volume fraction.
In figure 7, two types of the relative permittivity values are shown. The values obtained by
simulation and by analytical equations are of type εS and the experimental values are of type εT .
The effect of internal field modifications is not taken into considerations resulting in the deviation
of simulated and experimental values of dielectric constant. However, the dielectric constant values
obtained from analytical and numerical methods agree very well.
As the volume fraction becomes larger, the subsequent clamping of fibers by polymer has an
appreciable effect on the elastic and piezoelectric behavior; the elastic stiffness c33 is expected to
increase and piezoelectric strain constant e33 decreases. However, in the present work this kind of
clamping is not observed even up to 0.70 volume fraction. As the volume fraction increases, the
stiffening effect of adding more ceramic is mass loading (increased density) and this is responsible
for saturation effects observed in kt (figure 6) and Vl (figure 7).
The structural dimensions of a composite depend on the operating frequencies of the transducer.
13703-7
C.V. Madhusudhana Rao, G.Prasad
Figure 9. Variation of kt with aspect ratio.
The height of individual fiber is proportional to the ratio of the operating frequency and the
frequency constant of the ceramic. Hence, aspect ratio (height/width) is an important parameter
to be considered.
The dependence of kt on the aspect ratio of the piezoceramic fibers is also studied by simulation
for various volume fractions of piezoceramic fibers in the composite and the results are shown
graphically in figure 9.
Table 1. Comparison of Analytical and FEM results for a/l = 0.2.
ν C13
(GPa)
C33
(GPa)
e31
(C/m2)
e33
(C/m2)
ε33
(nF/m)
εr33 kt Vl
(m/s)
Z
(Mrayl)
0.1 A 4.90 13.2 –0.0364 1.854 0.796 89.9425 0.4963 3150 5.56
F 5.15 13.9 –0.0690 2.0142 0.869 98.1364 0.5014 3240 5.72
0.2 A 5.51 18.5 –0.0813 3.7016 1.55 175.6275 0.5679 3350 8.15
F 5.99 19.8 –0.1509 3.9747 1.68 190.3059 0.5675 3470 8.42
0.3 A 6.27 23.9 –0.1378 5.5401 2.31 261.2375 0.5972 3470 10.7
F 7.02 25.8 –0.2508 5.9402 2.50 282.8398 0.5947 3590 11.1
0.4 A 7.27 29.5 –0.2113 7.3655 3.07 346.7387 0.6118 3540 13.3
F 8.40 32.0 –0.3793 7.8511 3.31 374.0547 0.6065 3670 13.8
0.5 A 8.62 35.4 –0.3106 9.1709 3.82 432.0735 0.6189 3600 15.9
F 10.5 39.0 –0.5746 9.7955 4.14 467.9055 0.6105 3750 16.6
0.6 A 10.6 41.7 –0.4524 10.9435 4.58 517.1354 0.6207 3650 18.6
F 13.1 44.9 –0.7723 11.067 4.71 532.5207 0.6065 4370 16.4
0.7 A 13.5 48.9 –0.6713 12.6564 5.33 601.7016 0.6172 3700 21.3
F 18.6 53.5 –1.2282 12.4977 5.40 610.2610 0.5924 4680 17.6
A: Analytical values; F: Finite Element values.
From figure 9 it can be noted that, for a fixed volume fraction of the ceramic, the electro-
mechanical coupling coefficient (kt) decreases with the increase of the aspect ratio of the fiber.
Therefore, it can be noted that for higher values of kt, higher volume fraction of fiber material
is required and at the same time the aspect ratio of the fibers should be small. This behavior is
useful in optimizing the transducer performance. This result follows from the fact that the effective
elastic coefficient (c33) of the composite increased with the increase in the aspect ratio of the fibers,
although the other two quantities (c33 and ε33) remain almost unchanged.
To make sensitive transducers, an essential requirement is low Z and high kt. The present
calculations show that 1-3 composites can be superior in both respects; trade off being between
13703-8
Characterization of 1-3 piezoelectric polymer composites
minimizing impedance and maximizing coupling. This is because the lowering volume fraction
lowers not only Z but also the coupling.
For comparison of simulation results with analytical results for a/l = 0.2, the results are given
in table 1.
From the graphs, it can be observed that the analytical results obtained from effective medium
theory and the results obtained from FEM are in close agreement at all volume fractions. The
thickness mode electromechanical coupling coefficient (kt) is a very important transducer parameter
which we seek to be as high as possible. From figure 6 it can be observed that for ν = 0.6, kt
almost reaches its maximum possible value. That means we can achieve the same value of kt even
by reducing the content of piezoceramic by 40 percent. This effort results in lower density of the
composite material.
Another important requirement of a piezocomposite to be used in a transducer is the low
acoustic impedance (Z). From figure 5 it is noted that the acoustic impedance can be reduced by
reducing the volume content of piezoceramic (i.e. by reducing the volume fraction of piezoceramic,
ν) in the piezocomposite. At the same time, kt decreases with the decrease in ν, which we do
not need. Therefore, a tradeoff should be made between kt and Z for optimal performance of
transducer.
For better performance of a 1-3 composite transducer, there should be an effective transfer of
stress between the fiber material and the matrix material for which the surface area of interaction
should be made as large as possible [16]. This can be achieved by selecting long narrow fibers which
has lower aspect ratio (a/l) of fiber. We have also shown by the simulation results how the aspect
ratio of fibers affects the effective elastic coefficients and hence kt. These results prompt us to select
piezoceramic fibers with lower aspect ratio for higher kt values. By these trends the demand for
narrow fiber diameters increases which require high end technologies in the manufacturing process
of 1-3 piezocomposites [7, 17–21].
6. Conclusions
A framework has been worked out for simulation of essential ultrasonic transducer parameters
required for the composite transducer in medical or biological applications. The variations of these
parameters with aspect ratio are also worked out. The results are compared with experimental and
analytical values.
The effect of aspect ratio (a/l) on the effective properties of the 1-3 composite is not included
in the Smith’s analytical equation. However, when we analyzed the effect of (a/l) by simulation, it
turned out that the effective properties are more sensitive to (a/l) only at higher volume fractions.
It can also be noted that the relative permittivity is almost independent of (a/l) at all volume
fractions.
In the present work, piezoelectric fiber composites are modeled and analyzed by ANSYS soft-
ware by performing static analysis. The effective material properties of piezoceramic polymer com-
posites with 1-3 connectivity are determined. The dependence of the effective material properties
on the volume fraction of piezoceramic in the composite is analyzed by numerical results and then
the results are verified by using analytical equations. It is also noted that the fiber aspect ratio
(a/l) also effects the composite material properties at higher volume fractions. The results are
shown graphically for comparison. These results are useful in characterizing the piezocomposite
materials. The simulation procedure adopted is also useful in optimizing the material properties of
1-3 piezocomposites which are used in biomedical transducer applications. The simulation proce-
dure developed eliminates the trial and error method of optimization of piezocomposite transducer
materials. More importantly, optimization of aspect ratio for better kt is achieved which is not cov-
ered by the Smith’s model. Another advantage of simulation model is that different combinations
of piezoceramic fibers (with different a/l) and polymer matrix (piezo or non-piezo) materials can
be easily worked out without the use of any kind of laborious analytical procedures, the accuracy
being limited to the fineness of the model.
13703-9
C.V. Madhusudhana Rao, G.Prasad
References
1. Smith W.A. – In: Proc. of the 1989 IEEE Ulrrasonics Symposium, Montreal, Quebec, Canada, 3–6
October, 1989.
2. Gururaja T. R., Schulze W. A., Cross L. E., Newnham R. E., Auld B. A., Wang Y. J., IEEE T. Son.
Ultrason., 1985, 32, 481–498.
3. Kin Wing Kwok, Helen Lai Wah Chan, Chung Loong Choy, IEEE T. Ultrason. Ferr., 1999, 46, No. 3.
4. Newnham R. E., Skinner D. P., Cross L. E., Mater. Res. Bull., 1978, 13, 525–536.
5. Near C., Schmidt G., McNeal K., Gentilman R., SPIE, 1998, 3326, 323–331.
6. Safari A., J. Phys. III France, 1994, 4, 1129–1149.
7. Gebhardt S., Schönecker A., Steinhausen R., Hauke T., Seifert W., Beige H., Ferroelectrics, 2000,
241, 67–73.
8. Zhang D., Nelson L.J., Dent A., MacLennan D., Meggs C., Dolman G., Su B., Cochran S., Kirk K.J.,
Bowen C.R., Button T.W., Fine Scale Lead Zirconate Titanate (PZT) Fibres and Composite Struc-
tures. University of Bath, http://www.fmg.bham.ac.uk.
9. Iker F., André N., Proost J., Pardoen T., Raskin J.P., Fabrication and modeling of 3-D self-assembled
SOI MEMS using plasticity features. – In: Proc., 18th IEEE International Conference on Micro Electro
Mechanical Systems – MEMS 2005. Miami, USA, 2005, p. 459–462.
10. Nelli Silva E.C., Ono Fonseca J.S., Kikuchi N., Computational Mechanics, 1997, 19, 397–410.
11. Smith W.A., Shaulov A., Auld B.A., Tailoring the properties of composite piezoelectric materials for
medical ultrasonic transducers. – In: Proc., IEEE Ulfrason. Symp., 1985, p. 642–647.
12. Wah Chan H.L., Unsworth J., IEEE T. Ultrason. Ferr., 1989, 36, No. 4.
13. Smith W.A., Auld B.A., IEEE T. Ultrason. Ferr., 1991, 38, No. 1.
14. Steinhausen R., Charakterisierung und Modellierung piezoelektrischer 1-3 Komposite. Dissertation
Martin-Luther-University, Halle, Germany, 2002.
15. Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Guinovart R., Otero J.A., Bravo-Castillero J.,
Int. J. Solids Struct., 2005, 42, 5692–5714.
16. Wenwu Cao, Zhang Q.M., Cross L.E., J. Appl. Phys., 1992, 72 (12), 15.
17. Janas V.F., Safari A., J. Am. Ceram. Soc., 1995, 78, No. 11, 2945–2955.
18. Gebhardt S., Schoenecker A., Steinhausen R., Seifert W., Beige H., J. Eur. Ceram. Soc., 2003, 23,
153–159.
19. Glaubitt W., Watzka W., Scholz H., Sporn D., J. Sol-Gel Sci. Technol., 1997, 8, 29–33.
20. Steinhausen R., Hauke T., Beige H., Watzka W., Lange U., Sporn D., Gebhardt S., Schoenecker A.,
J. Eur. Ceram. Soc., 2001, 21, 1459–1462.
21. Klicker K.A., Piezoelectric composite with 3-1 connectivity for transducer applications. Ph.D. Thesis,
Solid State Science, The Pennsylvania State Univ., 1981.
Характеристика 1-3 п’єзоелектричних полiмерних
композитiв – числова i аналiтична процедура оцiнки вiбрацiй
мод товщини
С.В. Мадгусудгана Рао, Г. Прасад
Унiверситет Османiа, Гайдерабад 500 004 A.P, Iндiя
Бiомедичнi датчики широко використовують п’єзокерамiчнi полiмернi композити з 1-3 зв’язнiстю.
Дослiдження нацiлене на визначення ефективних матерiальних властивостей цих композитiв на мi-
крометричних масштабах методами комп’ютерного моделювання. Як було вже показано, об’ємна
доля п’єзокерамiки в композитi вiдiграє важливу роль у властивостях композитного матерiалу. По-
трiбно також вiдзначити вплив коефiцiєнта волокнистостi (a/l) на матерiальнi властивостi компо-
зита. Отже, метою є визначення ефективних матерiальних властивостей, використовуючи як ана-
лiтичнi методи, так i комп’ютерне моделювання, а саме, програму ANSYS, яка базується на методi
скiнченого елемента (FEM). У данiй роботi моделюються та аналiзуються п’єзоелектричнi волокнистi
композити, а отриманi результати перевiряються з використанням аналiтичних рiвнянь. Зроблено
важливi висновки щодо видiлення п’єзокомпозитних матерiалiв для застосування у датчиках.
Ключовi слова: п’єзоелектричнiсть, п’єзоелектричнi матерiали, електромеханiчнi ефекти,
п’єзоелектричнi датчики, методи скiнченого елементу
13703-10
Introduction
Piezoelectric 1-3 composites
Piezoelectric constitutive equations
Analytical equations
Numerical evaluation -- implementation of finite element method
Conclusions
|