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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:02 21
155802-6363-IJMME-IJENS © April 2015 IJENS
I J E N S
Tawakol A. Enab Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, P.O. 35516 Mansoura,
Egypt. Tel.: +201094684833, Fax: +20502244690, Emails: tenab@mans.edu.eg ; tewmkaln@yahoo.com
Abstract— The current study focuses on application of numerical
homogenization techniques to predict the effective
electromechanical properties of the periodic transversely
isotropic piezoelectric (PZT) cylindrical fiber composites.
ANSYS® finite element package used to develop the different
numerical representative volume element models of
unidirectional periodic composites made of piezoelectric fibers
surrounded by a soft non-piezoelectric polymeric matrix. The
formulation of the boundary conditions receives remarkable
interest to allow the simulation of all deformation modes coming
up from mechanical or electrical loadings or any arbitrary
combination of them. ANSYS Parametric Design Language
(APDL) used to generate all required constraint equations.
Effective electromechanical properties of unidirectional
piezoelectric cylindrical fiber composites with hexagonal fiber
arrangements calculated over a range of PZT fiber volume
fractions. Furthermore, for verification the homogenized elastic,
dielectric and piezoelectric properties compared to corresponding
analytical, numerical and experimental results reported in the
literatures, which demonstrate good correlations. The different
representative volume element models developed using APDL
scripts can provide a powerful tool for rapid calculation of
effective piezocomposite properties. Moreover, it can be applied
to composites with diverse inclusion geometries.
Index Term-- Finite element method (FEM); Micromechanics;
Numerical homogenization; Representative volume element;
Smart materials; Unidirectional piezoelectric composite.
I. INTRODUCTION
Composite materials represent an essential category of
current engineered materials since they have several
advantages over conventional materials. As a result of their
remarkable applications in our everyday uses, there is an urgent
need to predict the mechanical properties of these composites.
Techniques such as experimental studies and micromechanical
or macromechanical methods are used in this active research
field.
Furthermore, during the past few decades, numerous studies
on multifunctional materials and structures were carried out.
Piezoelectric material represents one of the most significant
multifunctional materials due to its electromechanical
interaction between the mechanical and the electrical energies
[1, 3]. Thus, the direct and indirect piezoelectric effects of these
materials are found in many useful applications such as sensors,
actuators, vibration and noise suppression, quartz watches,
medical instruments, harvesting kinetic energy from walking
pedestrians, etc.
But, as a result of bulk piezoelectric materials brittleness
nature, they have some limitations such as they are highly
susceptible to fracture and cannot be easily shaped to curved
surfaces. Therefore, piezoelectric composite materials present
an enhanced technological solution for many applications.
Actually, there are many types of piezocomposites depend on
the nature of connectivity. Piezoelectric composites with 0-3 or
1-3 connectivity are the most familiar piezocomposite types
and hence they were exclusively chosen for large area
fabrication. The 1-3 connectivity piezoelectric ceramic
fibers-polymer composites developed at MIT [4-6] using
piezoelectric ceramic fibers embedded in a passive
non-piezoelectric polymer. Therefore, the resultant composite
has better properties since it acquires the most advantageous
properties of each constituent material. Recently,
electromechanical sensors and actuators new applications
become possible due to the efficiency of piezoelectric
composites. Therefore, piezocomposites attract several
attentions to study its overall behavior, the local fields in its
constituent phases and its response under complex mechanical
and electrical loading conditions. Consequently, different
homogenization techniques used to describe the overall
coupled electromechanical behavior of piezoelectric
composites [7-9].
Using micromechanical techniques and through the analysis
of periodic unit cell models or representative volume elements
(RVEs) the overall coupled electromechanical behavior of
piezoelectric fiber composites can be obtained from the known
constituents properties. Since according to the
micromechanical approach, a homogeneous medium with
anisotropic properties can substitute the heterogeneous
structure of the composite. This approach has numerous
advantages such as obtaining the global properties of the
composites, studying various damage initiation and
propagation mechanisms, etc. [9]. Figure (1) demonstrates the
concept of representative volume element which fundamentally
reduces the original expensive analysis of heterogeneous
structures [10].
Evaluation of the Effective Electromechanical
Properties of Unidirectional Piezocomposites
Using Different Representative Volume
Elements
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:02 22
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I J E N S
Fig. 1. Structure analysis basic steps for heterogeneous microstructures using
the representative volume element concept [10].
Linear coupled electromechanical behavior of piezoelectric
composites can be predicted and simulated using different
methods. These methods include the basic analytical
techniques [11], mean field techniques [12-14], periodic
micro-field techniques (RVE or unit cell models) assisted by
finite element method (FEM) [15, 16], asymptotic
homogenization techniques [17, 18]. Recently, some
publications related to RVE models, which capture the overall
behavior of the composite appeared [7-9, 19-29].
Therefore, the current investigation aims to predict the
effective electromechanical properties of piezoelectric fiber
composites by calculating their complete elastic, dielectric and
piezoelectric tensors. Hence, the linear response of the
piezocomposite to any mechanical or electrical load or any
arbitrary combination of them can be determined using FEM
based micromechanical analysis method. This method used to
construct and analyze the different (representative volume
elements (RVEs) models of periodic unidirectional
piezoelectric cylindrical fiber composites having hexagonal
periodical distribution of fibers.
II. PIEZOELECTRICITY CONSTITUTIVE EQUATIONS
Linear constitutive equations of piezoelectricity can be
adequately used to model the coupled electromechanical
behavior of a piezoelectric material. Therefore, the application
of electric field, electrical displacements, mechanical stresses
or strains will result in a linear response of the piezoelectric
medium. These assumptions are well-matched with the
piezoelectric fibers, polymers, and piezocomposites used
nowadays [9]. Consequently, the electromechanical behavior
of piezoelectric material can be characterized using the
following linearly constitutive equations:
tyPermittiviricitypiezoelectDirect
ricitypiezoelectConverse
t
Elasticity
E
EeD
EeC
(1)
Here, the stress tensor (σ) and electric displacement vector
(D) are correlated to the strain vector () and the electric field
vector (E) by the stiffness matrix at constant electric field (CE),
the piezoelectric matrix (e) and the permittivity matrix at
constant strain (κ). Noting that, the matrix transpose is denoted
by the (t) superscript. The matrix form of the above equation
can be described as:
Eκe
e
D
tE
C (2)
The stiffness matrix (CE), the piezoelectric matrix (e), and
the dielectric matrix (κ) for transversely isotropic piezoelectric
solid can be simplified so that there remain only eleven
independent coefficients. Fortunately, the 1-3 piezocomposite
fabricated from transversely isotropic ceramic piezoelectric
fibers which surrounded by an isotropic polymeric matrix is
considered also as transversely isotropic piezoelectric medium
[7-9]. Accordingly, the above matrix form of constitutive
equation (2) can be demonstrated by the next form:
3
2
1
6
5
4
3
2
1
*
33
*
33
*
31
*
31
*
11
*
15
*
11
*
15
*
66
*
15
*
44
*
15
*
44
*
33
*
33
*
13
*
13
*
31
*
13
*
11
*
12
*
31
*
13
*
12
*
11
3
2
1
6
5
4
3
2
1
00000
0000000
0000000
00000000
0000000
0000000
00000
00000
00000
E
E
E
eee
e
e
C
eC
eC
eCCC
eCCC
eCCC
D
D
D
E
E
E
EEE
EEE
EEE
(3)
Thus, the coupled electromechanical problem general
variables found in the above matrix were substituted by the
effective coefficients of the homogenized material which
represented by elastic stiffness (CE*
ij), piezoelectric (e*ij) and
permittivity (κ*
ij) matrices. Moreover, the above equation also
comprises the average values of stress <i>, electric
displacement <Di>, strain <i> and electric field <Ei>
components. The aforementioned relationships designate the
corner stone for the supplementary considerations will be
applied to the representative volume element.
III. NUMERICAL HOMOGENIZATION TECHNIQUE
Homogenization technique basic idea can be simplified by
applying special load cases and suitable periodic boundary
conditions to the unit cell or representative volume element
(RVE) to determine the effective coefficients. Noting that, the
RVE is the basic structural unit of the material which at the
microscopic level comprises all required data can be employed
in the calculation of correct coefficients to describe the
macroscopic material behavior [30, 31]. In general, the
effective properties of unidirectional piezoelectric fiber
composites can be predicted by different analytical or
numerical homogenization techniques using the suitable RVE.
Analytical homogenization technique such as the asymptotic
homogenization method (AHM) usually based on RVEs with
Composite materials with
heterogeneous microstructure
Global analysis with
effective properties
Micromechanical
analysis of the RVE
Effective material
properties
Global responsesRecovery relations
Local displacements, strains, and
stress within the RVE
Global-local decomposition using
the concept of RVE
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:02 23
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I J E N S
simple cross-section inclusions such as circular or rectangular
which yield closed-form analytical expressions. But, on the
other hand, analytical homogenization techniques of complex
shape inclusions are difficult to carry out. Therefore, numerical
methods (e.g. finite element method FEM) developed to get
over the inclusions geometry and distribution restrictions.
Thus, both finite element and analytical homogenization
techniques using RVEs have advantages and disadvantages.
The analytical homogenization techniques are able to model
statistical distributions and consume less computing time than
finite element homogenization techniques. On the contrary, the
later are suitable for evaluating the effective coefficients of
composites having periodical distribution of fibers and more
complicated inclusions geometry. Moreover, they allow more
complex boundary conditions.
Generally, the fibers can have many types of arrangements
within the ordered fibrous composite. Hexagonal arrangement
regarded as one of the most important arrangement types. Thus,
the considered fibrous piezocomposite made up of in-line
hexagonal arrays of piezoelectric cylindrical fibers. Figure (2)
shows a schematic representation of hexagonal arrangements
of PZT fibers within the 1-3 piezoelectric composite and its
hexagonal RVE.
Fig. 2. Schematic representation of hexagonal arrangements of PZT fibers
within the piezoelectric composite and its hexagonal RVE.
A. Representative volume elements for numerical
homogenization techniques
Periodic 1-3 piezoelectric cylindrical fiber composites with
hexagonal arrangements have several representative volume
elements or unit cells geometrical forms as shown in figure
(3-a). The cylindrical fiber located at the center of rectangular
(R), hexagonal (H) and diamond (D1) representative volume
elements. While, fibers located at the corners of diamond RVE
(D1) and at the mid-edges of the rectangular RVE (R1). In
rectangular RVE (R2) the fibers located at corners and center. It
is clear that, any one of these periodical geometrical forms can
be employed as a RVE.
In piezoelectric composites with hexagonal arrangements,
rectangular RVEs (particularly R and R1) receive a lot of
attention (cited in [7-9, 28-32]). In view of the fact, this RVE
form facilitates the description and application of the necessary
periodic boundary conditions in the cartesian coordinate
system. But, in contrast, the Voronoi cell for hexagonal
arrangement is the hexagonal RVE (H) which represents the
optimum RVE choice as stated by Li [22]. In the current study,
the FEM micromechanical analysis method applied to the
different representative volume elements extracted from the
hexagonal arrangements of unidirectional piezoelectric fiber
composites. Therefore, our attention will be paid to study R1,
R2, D1, D2 and H RVEs. Consequently, the elastic stiffness
matrix (CE), piezoelectric matrix (e), and permittivity matrix
(κ) homogenized coefficients can be estimated for different
fiber volume fractions.
(a)
(b) R1-RVE
(c) R2-RVE
(d) H-RVE
(e) D1-RVE
(f) D2-RVE
Fig. 3. (a) Different periodical representative volume elements (RVEs) for
hexagonal arrangement [22]. (b-f) Coarse meshes of the 3D-RVEs models.
IV. FINITE ELEMENT MODELING
ANSYS software with coupled field elements used to
acquire the homogenized electromechanical properties of the
aforementioned RVEs. The general formulation governed by
the virtual work principle:
2
1
0)(t
tdtWU (4)
Hexagonal arrangement
Hexagonal RVE
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I J E N S
The internal energy (U) and external work (W) terms will
involve the piezoelectric contribution which represents the
major distinction between piezoelectric FEM and conventional
FEM formulations. Thus, we can associate the conventional
stress () - strain () behavior law with electric field (E) –
electric displacement (D) piezoelectric law. Another difference
is the electric potential () which considers as a supplementary
degree of freedom in addition to displacements (u).
Consequently, the coupled field elements in ANSYS used to
construct the three-dimensional models of the representative
volume elements. Figure (3. b-f) shows the 3D meshes of R1,
R2, H, D1 and D2 RVEs. Solid5, three dimensional
eight-noded elements used to develop the different RVEs finite
element models. Element formulation contains four degrees of
freedom (i.e. the three translations ux, uy, uz and the electric
potential ) at each nodal point. For easy and quick FEM
calculations an ANSYS Parametric Design Language (APDL)
macro developed for each RVE.
A. Periodic boundary conditions applied to the RVEs
The application of boundary conditions (BCs) is the
cornerstone of homogenization technique using finite element
method. Accordingly, formulation of the BCs should have a
notable interest to allow the simulation of all deformation
modes coming up from mechanical or electrical loadings or any
arbitrary combination of them. In consequence, periodic BCs
are applied to the 3D-RVEs models since any material can be
considered as repetitive patterns of the RVEs. Consequently,
the same deformation mode results for each RVE in the
piezocomposites. Moreover, the neighboring RVEs must have
no separation or overlap after deformation. Accordingly, the
application of periodicity conditions to the RVE surfaces can
be formulated as mentioned previously by [9, 17, 33] with the
following equation:
periodicuuxu iikiki
** , (5)
where <ik> represent the average strains and ui* represent
the local fluctuations (i.e. the displacement components
periodic part) on the RVE boundary surfaces. Noting that, the
local fluctuations depend on the applied loads and it is usually
unknown. The general expression given by equation (5) can be
reformulated to give a more explicit form of periodic BCs
which will be suitable for the developed RVEs models.
Taking the hexagonal RVE (H) shown in figure (4) as an
example, the displacements on the opposite boundary surfaces
are given by: ** , i
j
kik
j
ii
j
kik
j
i uxuuxu (6)
where the index „j+‟ and „j−‟ mean along the positive and the
negative Xj direction respectively on the RVE corresponding
surfaces (i.e. A1
+/A1
-, A2
+/A2
-, A3
+/A3
- and A4
+/A4
- , Fig. 4-a).
Noting that, due to RVE periodicity conditions, the local
fluctuations ui* are the same on each two opposing faces.
Therefore, the applied macroscopic strain conditions represent
the difference between the above two parts of equation (6), and
then they can be formulated as in the following equation.
j
kik
j
k
j
kik
j
i
j
i xxxuu )( (7)
Fig. 4. (a) Hexagonal representative volume. (b) Two nodes A et B are vis-à-vis
on two opposing faces.
In a similar manner, by using the applied macroscopic
electric field conditions, the electric potential periodic
boundary conditions can be written as: j
ki
j
k
j
ki
jj xExxE )( (8)
The opposite boundary surfaces of RVEs must have the same
meshes to facilitate the application of the aforementioned
periodic boundary conditions in the developed finite element
models. Therefore, a periodic boundary condition equations (7
and 8) are imposed for each pair of displacement and electrical
potential components at the two corresponding nodes with
vis-à-vis inplane positions on the two opposite boundary
surfaces (i.e. nodes A and B in Fig. 4-b, for example). Noting
that, remarkable interests are taken in imposing periodic
boundary condition equations of the vertices nodes of the
developed RVE models.
In view of the fact, the average properties of the piezoelectric
fiber composite are equal to the average properties of the RVE.
Therefore, the average stresses, strains, electric fields and
electrical displacements in the RVE can be written as:
V
ijij
V
ijij dVV
dVV
1
,1 (9)
V
ii
V
ii dVDV
DdVEV
E1
,1 (10)
where V is the periodic RVE volume.
V. RESULTS AND DISCUSSION
Since, the current investigation aims to predict the effective
electromechanical properties of piezoelectric composites with
hexagonal fiber arrangement using different RVEs models.
Moreover, these piezocomposites considered to be composed
of aligned transversely isotropic piezoelectric fiber embedded
in a soft non-piezoelectric isotropic polymer matrix (epoxy).
PZT-7A material was used in the developed models
calculations. Properties of PZT-7A and epoxy materials were
taken from references [13, 21] and are tabulated in table (1). All
homogenized coefficients (i.e. elastic stiffness, piezoelectric
and permittivity matrices) were calculated for twelve fiber
A2-
A1+
A2+ A3
+
A4+
A3−
A1−
A4−
A B
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:02 25
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I J E N S
volume fractions (vf ranges from 0.05 to 0.60 with a steps of
0.05).
The convergence of the finite element analysis for the
different RVE models was investigated in order to obtain
trustworthy results of the homogenized coefficients.
Accordingly, the modification of meshed RVEs by increasing
the meshes densities was carried out and continued until the
relative error becomes less than 1%.
Comparing the obtained effective coefficients of the
developed FEM models show that RVE models results are
overlapped with each other. But on the other hand, the effective
coefficients C11E*
, C66E*
, e15*, and 11
* results of the diamond
RVE (D1) have small variations at higher fiber volume
fractions as shown in figures (5 - 9).
Table I
Properties of piezoelectric PZT-7A and the dielectric matrix materials [13, 21*]
Properties of PZT-7A fiber
C11E
(GPa)
C12E
(GPa)
C13E
(GPa)
C33E
(GPa)
C44E
(GPa)
C66E
(GPa)
148 76.2 74.2 131 25.4 35.9*
e31
(C.m-2
)
e33
(C.m-2
)
e15
(C.m-2
) 11/0
33/0
-2.1 9.5 9.2 460 235
Properties of epoxy matrix
C11E
(GPa)
C12E
(GPa)
C13E
(GPa)
C33E
(GPa)
C44E
(GPa)
C66E
(GPa)
8 4.4 4.4 8 1.8 1.8*
e31
(C.m-2
)
e33
(C.m-2
)
e15
(C.m-2
) 11/0
33/0
0 0 0 4.2 4.2
Free space permittivity 0 = 8.85 (pC/N.m2).
Since, Mori-Tanaka (MT) mean field approach [13] is
commonly used successfully to various problems of mechanics
and physics of composite materials. Effective coefficients using
this approach can be defined as:
1
11
11*
).(...1
.).(.)..(
mfmesh
ff
mfmeshmf
f
m
CCCSIvIv
CCCSICCvCC
(11)
where: C* is the homogenized or effective coefficients tensor,
Cm
is the matrix tensor, Cf is the fiber tensor, v
f is the fiber
volume fraction, I is the identity tensor and finally Sesh
is the
Eshelby‟s tensor. Furthermore, many researches show good
agreement between Mori-Tanaka approach and the other
analytical, numerical and experimental techniques. Therefore,
another comparison carried out with this analytical approach
which shows, in general, good agreement between the results of
MT mean field approach and those of the developed RVE finite
element models. The homogenized coefficients C13E*
, C33E*
,
C44E*
, e31*, e33
*, and 33
* obtained by the two methods have a
very good match and nearly are indistinguishable. While on the
other hand, at higher fiber volume fractions, there is a little
differences between MT and FEM occur for the effective
coefficients C11E*
, C12E*
, C66E*
, e15*, and 11
* (Figs. 5 - 9). The
maximum difference does not exceed 6% for 60 vol. % of PZT
fiber and occurs for C66E*
effective coefficients (Fig. 7).
In addition, figure (10) presents a comparison between the
obtained FEM numerical results for the different RVEs and the
experimental results found in literatures [11, 13, 27] for
PZT-7A fiber composites. This figure shows plots of the
effective d33* versus the fiber volume fraction. It can be
observed that, the results of the developed RVE models using
the numerical homogenization technique have good agreement
the experimental data found in literatures. Moreover, it is
worthwhile to mention that the experimental results and those
obtained here follow the same trend.
Moreover, for more validation of the proposed
homogenization method, a comparison between the results of
the current study and that of the others has been carried out.
Table (2) illustrates the comparison of the effective properties
of a piezoelectric fiber composite with 60% PZT fiber volume
fraction. In the fore-mentioned table, numerical results
presented in [21], and the analytical solutions and FEM results
for hexagonal arrangement for the RVE of R1 type reported in
[9] were compared to those obtained using the different RVEs
developed in the present work. Noting that, the calculated
coefficients converted to the coefficients reported in [9, 21] to
obtain a more realistic comparison. Thus, the coefficients
conversion using the governing equations for constant electric
displacement may be given as the following:
*
11
*
15
*
15
*
33
*
33
*
33
*
33
*
13
*
31
*
66
*
66
*
11
*
15
*
44
*
44
*
33
*
33
*
33
*
33
*
33
*
33
*
13
*
13
*
13
*
33
*
13
*
12
*
12
*
33
*
13
*
11
*
11
*
33
*
33*
11
*
11
.
..
.
...
..
11
2
2
22
eh
eheh
CCeCC
eCCeeCC
eCCeCC
D
DD
DD
DD
DD
(12)
We can note that, the numerical predictions of the effective
coefficients using the different representative volume elements
show a good harmony with the other numerical and analytical
methods reported in references (see table 2). Moreover, the
homogenized coefficients for R1 and R2 RVEs are identical.
While there is a little discrepancies in the effective coefficient
C33D*
, but these discrepancies do not exceed 5% over the values
of numerical and analytical techniques found in the references.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:02 26
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I J E N S
Fig.5. C11
E* effective coefficient variation against PZT fiber volume fraction. Fig. 6. C12E* effective coefficient variation against PZT fiber volume fraction.
Fig. 7. C66
E* effective coefficient variation against PZT fiber volume fraction. Fig. 8. e15* effective coefficient variation against PZT fiber volume fraction.
Fig. 9. 11
* effective coefficient variation against PZT fiber volume fraction. Fig. 10. d33* effective coefficient variation against PZT fiber volume fraction.
4
8
12
16
20
24
0 10 20 30 40 50 60
C1
1E
*(G
Pa
)
PZT fiber Vf %
D1-RVE
D2-RVE
R1-RVE
R2-RVE
H-RVE
MT approach
0
2
4
6
8
10
12
0 10 20 30 40 50 60
C1
2E
*(G
Pa
)
PZT fiber Vf %
D1-RVE
D2-RVE
R1-RVE
R2-RVE
H-RVE
MT approach
0
1
2
3
4
5
6
7
0 10 20 30 40 50 60
C6
6E
*(G
Pa
)
PZT fiber Vf %
D1-RVE
D2-RVE
R1-RVE
R2-RVE
H-RVE
MT approach
0.00
0.01
0.02
0.03
0.04
0.05
0 10 20 30 40 50 60
e1
5*
(C/m
2)
PZT fiber Vf %
D1-RVE
D2-RVE
R1-RVE
R2-RVE
H-RVE
MT approach
0
4
8
12
16
20
0 10 20 30 40 50 60
k1
1*
/ k
0
PZT fiber Vf %
D1-RVE
D2-RVE
R1-RVE
R2-RVE
H-RVE
MT approach
0
25
50
75
100
125
150
175
200
0 10 20 30 40 50 60 70 80 90 100
d3
3*
(pC
/N)
PZT fiber Vf %
D1-RVE
D2-RVE
R1-RVE
R2-RVE
H-RVE
MT approach
Experimental
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I J E N S
Table II
Comparison of the piezoelectric homogenized coefficients for piezocomposite has 60% PZT fiber volume fraction.
FEM-
HEX.
D1-RVE
FEM-
HEX.
D2-RVE
FEM-
HEX.
H-RVE
FEM-
HEX.
R1-RVE
FEM-
HEX.
R2-RVE
Berger et.
al.
FEMHEX.
[9]
R1-RVE
Berger
et. al.
AHM-
HEX.
[9]
Pettermann
– Suresh
[9,21]
SCS
Levin
[9]
HS bounds
[9]
C11D* (GPa) 22.689 22.329 22.313 22.357 22.357 22.40 22.40 22.41 21.84 24.9 / 28.7
C12D* (GPa) 10.292 10.420 10.409 10.447 10.447 10.50 10.51 10.51 10.99 5.0 / 12.0
C13D* (GPa) 10.148 10.083 10.076 10.099 10.099 10.52 10.53 10.53 10.51 6.12 / 16.5
C33D* (GPa) 90.494 90.483 90.476 90.538 90.538 86.89 86.91 86.91 86.90 79.0 / 87.8
C44D* (GPa) 6.299 6.341 6.325 6.361 6.361 6.31 6.3483 6.34 6.307 6.40 / 7.67
C66D* (GPa) 5.792 5.976 5.952 6.017 6.017 5.95 5.943 5.95 5.424 4.37 / 4.92
11* (GVm/C) 6.564 6.796 6.812 6.732 6.732 6.807 6.619 6.809 6.859 2.54 / 6.73
* (GVm/C) 0.779 0.779 0.779 0.778 0.778 0.7799 0.7797 0.780 0.7796 0.742 / 0.951
h31* (GV/m) -0.158 -0.157 -0.156 -0.157 -0.157 −0.1498 −0.1498 −0.150 -0.1493 -1.03 /0.719
h33* (GV/m) 4.982 4.982 4.982 4.982 4.982 5.039 5.039 5.039 5.039 3.63 / 5.85
h51* (GV/m) 0.299 0.290 0.289 0.291 0.291 0.2873 0.3045 0.289 0.2844 -1.92 / 2.67
Fig. 11. Deformations of different RVEs for (a) <1> =1; applied u1 displacement, (b) <2> =1; applied u2 displacement, (c) <3> = 1; applied u3
displacement, and (d) <E3> =1; applied electric potential.
(a) <x> =1
(b) <y> =1
(c) <z> = 1
(d) <Ez> =1
R1-RVE R2-RVE D1-RVE D2-RVE H-RVE
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:02 28
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I J E N S
Figure (11) shows the deformations and electric potential
distributions for the different RVEs when applying the
constraints to obtain the homogenized coefficients. For
example, by applying =1 while keeping other components
of the macroscopic applied strain equal to zero (refer to eq. 3),
we can deduce directly homogenized constants C11*, C12
*, C13
*
and e31*.
VI. CONCLUSIONS
Finite element numerical models for different representative
volume elements developed to calculate the effective properties
of 1-3 connectivity piezocomposites with hexagonal
arrangement. General purpose finite element method software
(ANSYS) with coupled field elements was employed in the
development of the different RVEs models using ANSYS
Parametric Design Language (APDL) for easy and fast
calculations. These aforementioned RVEs used to predict all
effective coefficients for different volume fractions which
facilitate the study of the influence of piezoelectric fiber
volume fraction on the homogenized coefficients of the
piezocomposite. All electromechanical effective coefficients
(i.e. elastic stiffness, piezoelectric and permittivity matrices
parameters) have been predicted for the different forms of
representative volume elements. Good agreement between the
predicted and experimental, numerical and analytical
coefficients found in literatures which therefore support the
usability of the developed models in the current study.
Moreover, the developed models reduce the long-lasting
manual work and can be extended to predict the homogenized
coefficients of piezocomposites with arbitrary fiber
arrangement and/or with reinforcement complex geometries.
Also, it can be extended to include other coefficients such as
those for pyroelectric or any other material properties
combinations.
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Tawakol A. ENAB graduated from Faculty of
Engineering, Mansoura University, Egypt and
received the Ph.D. degree from Savoie
University, Chambery, France. Currently, he is
an Assistant Professor at Production
Engineering and Mechanical Design
Department, Faculty of Engineering, Mansoura
University, Mansoura, Egypt. His research
interests comprise: the mechanical behavior of
advanced composite materials, piezoelectric
composites, functionally graded materials and
biomaterials.