evaluation of the effective electromechanical properties ... · ansys parametric design language...

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: [email protected] ; [email protected]

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

mailto:[email protected]

mailto:[email protected]



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



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



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



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.



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



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



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.

REFERENCES [1] J.-S. Park, J.-H. Kim, “Analytical development of single crystal Macro

Fiber Composite actuators for active twist rotor blades”, Smart Materials and Structures, Vol. 14, No. 4, pp. 745-753, 2005.

[2] M. Li, T. C. Lim, Y. H. Guan, W. S. Jr. Shepard, “Actuator design and

experimental validation for active gearbox vibration control”, Smart Materials and Structures, Vol. 15, No. 1, Technical note, pp. N1-N6,

2006.

[3] H. A. Sodano, J. Lloyd, D. J. Inman, “An experimental comparison between several active composite actuators for power generation”, Smart

Materials and Structures, Vol. 15, No. 5, pp. 1211-1216, 2006.

[4] A. A. Bent, “Active fiber composites for structural actuation”, PhD Dissertation, Department of Aeronautics and Astronautics, Massachusetts

Institute of Technology, 1997.

[5] A. A. Bent, N. W. Hagood, “Piezoelectric fiber composites with interdigitated electrodes”, Journal of intelligent material systems and

structures, Vol. 8, pp. 903-919, November 1997.

[6] N. W. Hagood, A. A. Bent, “Development of piezoelectrical fiber composites for structural actuation”, In: Proc 34th AIAA Structures,

Structural Dynamics and Materials Conference, La Jolla, CA, Part 6, pp. 3625–38, 1993.

[7] H. Berger, S. Kari, U. Gabbert, R. Rodriguez-Ramos, J. Bravo-Castillero,

R. Guinovart-Diaz, “A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric

fibre composites”, Materials Science and Engineering A, Vol. 412, pp.

53–60, 2005.


R. Guinovart-Diaz., “Calculation of effective coefficients for

piezoelectric fiber composites based on a general numerical

homogenization technique”, Composite Structures, Vol. 71, pp. 397–400, 2005.


R. Guinovart-Diaz, F. J. Sabina, G. A. Maugin, “Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of

effective properties”, Smart Materials and Structures, Vol. 15, No. 2, pp.

451-458, 2006.

[10] W. Yu, T. Tang, “Variational asymptotic method for unit cell

homogenization of periodically heterogeneous materials”, International

Journal of Solids and Structures, Vol. 44, pp. 3738–3755, 2007.

[11] H. L. W. Chan, J. Unsworth, “Simple model for piezoelectric

ceramic/polymer 1-3 composites used in ultrasonic transducer

applications”, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 36, pp. 434–441, 1989.

[12] Y. Benveniste, “Universal relations in piezoelectric composites with

eigenstress and polarization fields. Part I: Binary media - local fields and effective behavior”, Journal Applied Mechanics, Vol. 60, pp. 265–269,

1993.

[13] M. L. Dunn, M. Taya, “Micromechanics predictions of the effective

electroelastic moduli of piezoelectric composites”, International Journal

of Solids and Structures, Vol. 30, No. 2, pp. 161–175, 1993.

[14] B. Wang, “Three-dimensional analysis of an ellipsoidal inclusion in a piezoelectric material”, International Journal of Solids and Structures,

Vol. 29, pp. 293–308, 1992.

[15] P. Gaudenzi, “On the electromechanical response of active composite materials with piezoelectric inclusions”, Computers and Structures, Vol.

65 157–68, 1997.

[16] H. J. Bohm, “Numerical investigation of microplasticity effects in

unidirectional long fiber reinforced metal matrix composites”, Modeling

and Simulation in Materials Science and Engineering, Vol. 1, pp. 649–671, 1993.

[17] P. Suquet, “Elements of homogenization theory for inelastic solid

mechanics Homogenization Techniques in Composite Media”, Ed. Sanchez-Palencia E. and Zaoui A., Springer publications, pp. 194–278,

1987.

[18] N. S. Bakhvalov, G. P. Panasenko, “Homogenization Averaging

Processes in Periodic Media”, Kluwer publications, 1989.

[19] A. Agbossou, C. Richard, Y. Vigier, “Segmented piezoelectric fiber

composite for vibration control: fabricating and modeling of electromechanical properties”, Composites Science and Technology, Vol.

63, pp. 871-881, 2003.

[20] H. Sun, S. Di, N. Zhang, C. Wu, “Micromechanics of composite materials using multivariable finite element method and homogenization theory”,

International Journal of Solids and Structures, Vol. 38, pp. 3007–3020,

2001.

[21] H. E. Pettermann, S. Suresh, “A comprehensive unit cell model: a study

of coupled effects in piezoelectric 1-3 composites”, International Journal

of Solids and Structures, Vol. 37, pp. 5447–5464, 2000.

[22] S. Li, “General unit cells for micromechanical analyses of unidirectional

composites”, Composites Part A, Vol. 32, pp. 815–826, 2000.

[23] J. Pastor, “Homogenization of linear piezoelectric media”, Mechanic Res. Comm., Vol. 24, No. 2, pp. 145-150, 1997.

[24] P. Tan, L. Tong, “Micro-electromechanics models for

piezoelectric-fiber-reinforced composite materials”, Comput. Sci. Technol., Vol. 61, pp. 759–769, 2001.

[25] T. A. Enab, “Numerical homogenization technique for calculating

effective material coefficients of 1-3 piezocomposites”, Proceedings of the 4th Assuit University International Conference on Mechanical

Engineering Advanced Technology for Industrial Production

(MEATIP4), Mechanical Engineering Department, Faculty of Engineering, Assuit University, Assuit, Egypt, 12-14 December 2006, pp.

153-160.

[26] T. A. Enab, “Numerical investigation on effective electromechanical properties of unidirectional piezoelectric composites using different unit

cells models”, Proceedings of Al-Azhar Engineering Ninth International

Conference (AEIC2007), Faculty of Engineering, Al-Azhar University, Nasr City, Cairo, Egypt, 12-14 April 2007, and Al-Azhar University

Engineering Journal (JAUES), Vol. 2, No. 4, April 2007, pp. 312-321.



I J E N S

[27] Y. Koutsawa, F. Biscani, S. Belouettar, H. Nasser, E. Carrera,

“Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials”,

Composite Structures, Vol. 92, pp. 964–972, 2010.

[28] M. Würkner, H. Berger, U. Gabbert, “Numerical homogenization of composite structures with rhombic fiber arrangements”, PAMM Special

Issue: 82nd Annual Meeting of the International Association of Applied

Mathematics and Mechanics (GAMM), Volume 11, Issue 1, pp. 559–560, December 2011.

[29] M. Würkner, H. Berger, U. Gabbert, “Numerical study of effective elastic

properties of fiber reinforced composites with rhombic cell arrangements and imperfect interface”, International Journal of Engineering Science,

Vol. 63, pp. 1-9, February 2013.

[30] N. Bohn, U. Gabbert, “Optimization of piezoelectric fiber distribution in composite smart materials”, PAMM. Proc. Appl. Math. Mech., Vol . 4,

pp. 302-303, 2004.

[31] S. Kari, H. Berger, U. Gabbert, “Micro-macro modeling of piezoelectric fiber composites”, PAMM. Proc. Appl. Math. Mech., Vol . 4, pp.

330-331, 2004.

[32] A. R. Aguiar, J. B. Castillero, R. R. Ramos, U. P. da Silva, “Effective

Electromechanical Properties of 622 Piezoelectric Medium With

Unidirectional Cylindrical Holes”, Journal of Applied Mechanics, ASME

transactions, Vol. 80, Issue 5, Paper No: JAM-12-1316, Septemper 2013.

[33] Z. Xia, Y. Zhang, F. Ellyin, “A unified periodical boundary conditions for

representative volume elements of composites and applications”,

International Journal of Solids and Structures, Vol. 40, pp. 1907-1921, 2003.

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.

evaluation of the effective electromechanical properties ... · ansys parametric design language...

Documents