Investigation of electromechanical properties of piezoelectric structural fiber composites with micromechanics analysis and finite element modeling

Mechanics of Materials 53 (2012) 29–46

Contents lists available at SciVerse ScienceDirect

Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat

Investigation of electromechanical properties of piezoelectric structural fiber composites with micromechanics analysis and finite element modeling Qingli Dai ⇑, Kenny Ng Dept. of Civil and Environmental Engineering, Michigan Technological University, Houghton, MI 49931, United States

a r t i c l e

i n f o

Article history: Received 14 June 2011 Received in revised form 30 March 2012 Available online 25 May 2012 Keywords: Piezoelectric structural fiber composites Electromechanical properties Finite element modeling Mori–Tanaka approach Rule of Mixtures

a b s t r a c t This paper investigates the electromechanical properties of piezoelectric structural fiber (PSF) composites with the combined micromechanics analysis and finite element modeling. The active piezoelectric materials are widely used due to their high stiffness, voltage-dependent actuation capability, and broadband electro-mechanical interactions. However, the fragile nature of piezoceramics limits their sensing and actuating applications. In this study, the active PSF composites were made by deploying the longitudinally poled PSFs into a polymer matrix. The PSF itself consists a silicon carbide (SiC) or carbon core fiber as reinforcement to the fragile piezoceramic shell. To predict the electromechanical properties of PSF composites, the micromechanics analysis was firstly conducted with the dilute approximation model and the Mori–Tanaka approach. The extended Rule of Mixtures was also applied to accurately predict the transverse properties by considering the effects of microstructure including inclusion sizes and geometries. The piezoelectric finite element (FE) modeling was developed with the ABAQUS software to predict the detailed mechanical and electrical field distribution within a representative volume element (RVE) of PSF composites. The simulated energy or deformation under imposed specific boundary conditions was used to calculate each individual property with constitutive laws. The comparison between micromechanical analysis and finite element modeling indicates the combination of the dilute approximation model, the Mori–Tanaka approach and the extended Rule of Mixtures can favorably predict the electromechanical properties of three-phase PSF composites. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Over the past few decades, the active or multifunctional materials have been significantly advanced for the applications in sensing/actuating, vibration damping, damage detection etc. The active piezoelectric materials are widely used due to their strong stiffness, voltage-dependent actuation authority, and electro-mechanical interaction over broad frequencies. However, the monolithic piezoceramic materials such as lead-based ceramics are brittle by nature. ⇑ Corresponding author. E-mail addresses: [email protected] (Q. Dai), [email protected] (K. Ng). 0167-6636/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmat.2012.04.014

The fragile property makes them vulnerable to accidental breakage during operations, and difficult to apply to curved surfaces and harsh environments with reduced durability (Lin and Sodano, 2009). To overcome the fragile nature of piezoceramics, piezoelectric fiber composites (PFCs) were fabricated by embedding a fibrous form of piezoceramics into a polymer matrix (Bent, 1994). Different types of PFCs have been reported by different researchers. Bent (1997) and Bent and Hagood (1997) proposed the first prototype PFC, also called as Active Fiber Composite (AFC). The lead zirconium titanate oxide (PZT) fibers were unidirectionally deployed in the polymer matrix. Following this work, NASA Langley

30

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Research Center developed macro fiber composite (MFC) to address the limitation in the electric field of the circular cross-section of AFC fibers. Rectangular shape fibers were applied in the MFC. Both AFC and MFC use a separate interdigitated electrode pattern that is bonded to the fibers’ transverse surface which can make it difficult to fabricate into the polymer matrix. One type of hollow fiber composite (HFC) was proposed by Brei and Cannon (2004) to allow the electric field to be applied through the thickness of the hollow fiber. Thus the impedance of the material and the required actuation voltage were reduced. Recent research identified the potential of piezoelectric structural fibers (PSFs) for improving structural multifunction and also overcoming the fragile nature of monolithic piezoceramics. Metal core PFCs were developed to overcome the brittleness of the HFC by coating a metal fiber (typical platinum fiber) with PZT (Takagi et al., 2001). The metal core can reinforce the composite and serve as one electrode. However, the mismatch in the ductility and the coefficient of thermal expansion of the metal conductor and piezoceramic coating makes the coating prone to cracking under thermo-mechanical process. To address these issues, Lin and Sodano (2008, 2009) recently developed the active structural fiber (ASF) by applying a thin PZT film over the surface of carbon or SiC fibers to strengthen the composites. And the conductive core fibers also serve as internal electrodes. However, the difficulty in applying or collecting the electric fields along the piezoelectric shell of the transversely poled PSFs makes it unsuitable for autonomous sensing or actuating applications. The advance of multifunctional composite materials requires the development of the coupled models to predict the interactions between mechanical responses and electrical fields. Eshelby’s classical formulation has been widely applied to study the stress–strain field in the composite materials (Eshelby, 1957). Researchers have extended Eshelby’s solution to predict the electromechanical properties of piezoelectric composites. Dunn and Taya (1993a,b) and Dunn and Wienecke (1996, 1997) formulated the dilute, self-consistent, Mori–Tanaka and differential micromechanics models to predict the coupled electroelastic behaviors of piezoelectrical composite materials for different inclusion geometries. The results indicated that the Mori–Tanaka micromechanical schemes have the closest prediction with the experimental data. Fakri et al. (2003) also formulated the interaction tensors for the self-consistent model, Mori–Tanaka and Dilute approaches to predict electroelastic properties with various reinforcement orientations. The study concluded that the Mori–Tanaka approach has the best agreement with the experimental data. The comparison study between the micromechanics prediction of the electroelastic properties and finite element analysis of piezoelectric composites were conducted by Odegard (2004) and Huang and Kuo (1996). In addition, Kar-Gupta and Venkatesh (2007) developed a micromechanics model to capture the electromechanical interactions of a 1–3 piezoelectric composite system where both the matrix and fiber phases are active materials. And a micromechanical model was recently presented to estimate the electroelastic behavior of the piezoelectric composites with coated reinforcements by Dinzart and Sabar (2009).

Even though many micromechanics models have been developed to study the elastic properties of multi-phase composite materials in the past decade, only few micromechanics researches for understanding the electroelastic properties of multi-phase active materials have been reported. In order to predict the elastic properties of the composites with multi-layered inclusions, the double inclusion model proposed by Hori and Nemat-Nasser (1994) extends the general dilute approximation method to calculate the average stress–strain field. Dunn and Ledbetter (1995) extended the double inclusion theory to predict the elastic moduli of three-phase composites consisting of a matrix and two-phase reinforcement particles. The theoretical predictions have good agreement with experimental data for mullite/alumina particle reinforced aluminum composites. Furthermore, Li (1999, 2000) employed the multi-inclusion theories to study the thermoelastic moduli and magnetoelectroelastic interactions within composites. In this paper, the combined models with the dilute approximation model, the Mori–Tanaka approach and the extended Rule of Mixture were proposed to predict the electromechanical properties of PSF composites. The objective of this study is to formulate the combined micromechanics models and to compare the model prediction with the finite element analysis of PSF composites. The Section 2 briefly demonstrates the internal structure and fabrication procedure of PSFs and PSF composites. The Section 3 formulates the combined micromechanics modeling with the dilute approximate model, the Mori–Tanaka approach and the extended Rule of Mixtures to predict the electromechanical properties of PSF composites. The finite element (FE) model and property analysis methods are illustrated in the Section 4. Finally, the comparison results between micromechanics model prediction and FE analysis are presented in the last section. Overall, the comparison results show that the combined micromechanics models can favorably predict the electromechanical properties of PSF composites.

2. Piezoelectric structural fiber (PSF) composites The active PSF composites are made by unidirectionally deploying the longitudinally poled PSFs into a polymer matrix. The presented PSFs are fabricated by coating piezoceramics onto a carbon/silicon carbide (SiC) core fiber to enhance the mechanical properties of monolithic piezoelectric materials as shown in Fig. 1. Fig. 1(a) shows the internal structure of a PSF: core carbon/SiC fiber serves as the reinforcement to the fragile piezoceramic shell. The surrounding media is the epoxy polymer matrix. The aspect ratio, a of the PSF is defined as the value of shell thickness, t divided by the outer radius, ro, as shown in Fig. 1(a). The PSF was longitudinally poled along the fiber direction. The active composite laminate with deployed PSFs is demonstrated in Fig. 1(b). The electrical inputs/outputs are delivered through a separate built-in electrode layer. The volume fraction of the PSFs is the volume ratio of fibers with the whole laminate. Under the vibration deformation, the electrical output of this laminate transducer will be

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

31

Fig. 1. Demonstration of piezoelectric structural fiber (PSF) composites, (a) structure of one PSF and (b) PSF laminate with built-in electrodes.

used for multifunctional applications such as power harvest and damping control. In the reverse manner, the actuating function of this laminate transducer can be generated by imposing the electrical voltage input. The lab fabrication of PSF composites was conducted with the eletrophoretic deposition (EPD), fiber sintering and polarization processes. The EPD process is to coat piezoceramic thin or thick films onto the core fibers. Following the EPD process, the coated fibers were sintered in a tube furnace at a high temperature setting under the nitrogen gas atmosphere. During fiber sintering, the coated layer also generated the phase transition from small cubic particles to a crystal structure with improved density. To form the input/output electrodes, the silver paint (epoxy-polymer) is coated at two ends. The PSFs were longitudinally polarized under a high DC electrical field between the top and bottom silver electrodes and at its curie temperature (120 °C) (Lu et al., 1993). The poled PSFs were uniformly assembled between two aluminum cure plate molds. The matrix materials were introduced and cured between the molds in a vacuum setting. The input/output copper electrodes were pressed as electrodes for the sensing and actuating applications. 3. Combined micromechanics model for PSF composites This section presents the linear electromechanical constitutive equations for the piezoelectric materials. The

combined micromechanics model including the dilute approximate model, the Mori–Tanaka approach and the extended Rule of Mixtures is also described for predicting the effective electro-mechanical properties of PSF composites. 3.1. Electro-mechanical constitutive behavior for piezoelectric ceramic In this paper, the piezoelectric materials are assumed to have linear electromechanical behaviors. Under low electric fields or mechanical loadings, the piezoelectric materials have linear coupled electro-mechanical responses (IEEE, 1988; Reza Moheimani and Fleming, 2006). Depending on the applications, the piezoelectric materials can be used as sensors or actuators. For the sensing mode, the linear electro-mechanical constitutive equations that describe the coupled interaction between the mechanical and electrical variables are given as followings (Dai and Ng, 2010):

eij ¼ Sijmn rmn þ dnij En

ð1Þ

Di ¼ dimn emn þ jTin En

ð2Þ

where e, r, E, and D are the strain, stress, electric field, and electric displacement tensors, respectively, and the S, d, and jT are elastic compliance tensor (at a constant electric field), piezoelectric field-strain tensor (in a constant stress or electric field), and dielectric tensor (at a constant stress), respectively. These equations use the conventional indicial

32

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

notation, and the repeated subscripts are summed over the range of i; j; m; n ¼ 1; 2; 3. The strain–displacement relation and the electric field-potential equation are given as:

1 2

eij ¼ ðui;j þ uj;i Þ and cij ¼ ui;j þ uj;i ;

ð3Þ

En ¼ u;n

ð4Þ

where e, U, and E are the strain, displacement, and electric field respectively, and u is the electric potential. While for the actuating mode, the linear electromechanical constitutive equations that describe the coupled interaction between the mechanical and electrical variables are presented as followings (Reza Moheimani and Fleming, 2006):

rij ¼ C ijmn emn  enij En

ð5Þ

Di ¼ eimn emn þ jSin En

ð6Þ

where C, e, and jS are the elastic modulus tensor (at a constant electric field), piezoelectric field-stress tensor (in a constant strain or electric field), and dielectric tensor (at a constant strain), respectively. In order to model the PSF composites with piezoelectric shells as shown in Fig. 1(a), it is convenient to combine the mechanical and electrical variables in Eqs. (5) and (6) to one single constitutive equation. The notation is identical to the conventional indicial notation except that the lower case subscripts are in the range of 1–3, while the capitalized subscripts are in the range of 1–4 and repeated capitalized subscripts summed over 1–4. With these notations, the linear piezoelectric constitutive behavior can be expressed as the following (Lin and Sodano, 2010; Odegard, 2004):

PiJ ¼ EEiJMn ZMn

ð11Þ ! 1 ð22Þ ! 2 ð33Þ ! 3 ð23Þ ! 4 ð13Þ !5

ð11Þ

ð12Þ ! 6

ð14Þ ! 7 ð24Þ ! 8 ð34Þ ! 9

ð12Þ

where in EEiJMn , the first pair of indices describes the row in the 9  9 matrix and the second pair represents the column by following the mapping Eqs. (11) and (12). The Eq. (11) uses the well-known Voigt elastic notations while Eq. (12) represents Voigt two-index notations to include the piezoelectric and dielectric constants in the. Hence, for an orthotropic piezoelectric material, the extended constitutive equation with 9  9 matrix can be expressed as the followings: 3 2 C 11 r11 6r 7 6C 6 22 7 6 12 7 6 6 6 r33 7 6 C 13 7 6 6 7 6 6 6 r23 7 6 0 7 6 6 6 r13 7 ¼ 6 0 7 6 6 7 6 6 6 r12 7 6 0 7 6 6 6 D1 7 6 0 7 6 6 7 6 6 4 D2 5 4 0 D3 e31 2 2

C 12 C 22 C 23 0 0 0 0 0 e32 3

C 13 C 23 C 33 0 0 0 0 0 e33

0 0 0 C 44 0 0 0 e15 0

0 0 0 0 C 55 0 e15 0 0

0 0 0 0 0 C 66 0 0 0

0 0 0 0 e15 0

j1 0 0

0 0 0 e15 0 0 0

j2 0

3 e31 e32 7 7 7 e33 7 7 7 0 7 7 0 7 7 7 0 7 7 0 7 7 7 0 5

j3

e11 6e 7 6 22 7 7 6 6 e33 7 7 6 7 6 6 c23 7 7 6 7  6 6 c13 7 7 6 6 c12 7 7 6 6 E1 7 7 6 7 6 4 E2 5 E3

ð7Þ

ð13Þ

and the stress and electric displacement can be combined as:

In this equation, the 9  9 electro-elastic modulus matrix can be represented with the index p and q. The index p and q range from 1 to 9. Thus, the electro-elastic modulus matrix can also be divided into four sub-matrices as:

 PiJ ¼

rij ; J ¼ 1; 2; 3; Di ;

J¼4

ð8Þ

and similarly, the strain and electric field can be represented as:

( ZMn ¼

emn ; M ¼ 1; 2; 3; En ;

M ¼ 4:

ð9Þ

The electro-elastic modulus tensor can then be expressed as:

EEiJMn ¼

8 C ijmn ; J; M ¼ 1; 2; 3; > > > > > > > < enij ; J ¼ 1; 2; 3; M ¼ 4 > > eimn ; > > > > > : jin ;

ð10Þ

J ¼ 4; M ¼ 1; 2; 3; J; M ¼ 4

In the Eqs. (7)–(10), the key variables PiJ and ZMn are represented with the 9  1 column vectors, and the EEiJMn is a diagonally symmetric matrix with the dimension of 9  9. In these notations, the following mapping of adjacent indices, e.g. (iJ) = (Ji) and (Mn) = (nM) for J and M – 4, are utilized (Dunn and Taya, 1993b):

EEpq

8 C pq ; > > > < e ; kq ¼ > ekq ; > > : jpq ;

p; q ¼ 1—6; k ¼ 1—6; q ¼ 7—9 k ¼ 7—9; q ¼ 1—6;

ð14Þ

p ¼ 7—9; q ¼ 7—9

where C, e and j are the elastic modulus, the piezoelectric field-stress tensor and the dielectric tensors. The effective piezoelectric field-strain tensor d is widely used to evaluate the piezoelectric coupling properties. Based on the IEEE standard (IEEE, 1988), the effective piezoelectric fieldstrain tensor d can be determined from the following relationship:

ekq ¼ dkp C pq

ð15Þ

with elastic modulus tensor C and the piezoelectric fieldstress tensor e. 3.2. Combined micromechanics model for PSF composites In this study, the PSF composites are fabricated by deploying the longitudinally poled PSFs into a polymer ma-

33

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

trix. The PSF itself consists a silicon carbide (SiC) or carbon core fiber as reinforcement to the fragile piezoceramic shell. The cross-section structure of a representative volume element (RVE) of a PSF composite is shown in Fig. 2. In order to apply the active PSF composites, it is essential to predict the effective electro-mechanical property of these composites. It is assumed that the interfaces among N phases have perfect interfacial bonding in the PSF composites. The volume averaged piezoelectric fields of these multiphase active materials can be then expressed as (Lin and Sodano, 2010; Odegard, 2004):

P¼

N X cr Pr

ð16Þ

r¼1

¼ Z

N X r cr Z

ð17Þ

r¼1

where the overbar denotes the volume averaged field, cr is the volume fraction of rth phase, and the subscript r ¼ 1; 2; 3 indicates each individual phase: polymer matrix, piezoelectric material (such as PZT-5H), and core fiber as shown in Fig. 2. And the polymer matrix is the reference medium as phase 1. With these volume averaged fields, the constitutive equation (Eq. (7)) for a PSF composite can be expressed as:

 P ¼ EEZ

ð18Þ

 r ) of the and the volume averaged strain and electric field (Z rth phase, used in Eq. (17), can be written with the average  in the following equation: composite field Z

 r ¼ Ar Z  Z

1 1 Adil r ¼ ½I þ Sr EE1 ðEEr  EE1 Þ

ð20Þ

r¼1

where I is the fourth order identity tensor. In order to determine the concentration tensor Ar, the dilute approximation model and the Mori–Tanaka approach are used in this study. Another widely-used self-consistent method yields the implicit matrix equation for determining the concentration tensor Ar. So the numerical integration need be employed, and thus leads to slow and complicated calculations for the electro-elastic properties (Dunn and Taya, 1993a).

3.2.2. Mori–Tanaka approach The Mori–Tanaka approach (Mori and Tanaka, 1973) has been applied to solve many problems related to physical properties of composite materials. The basic idea of the Mori–Tanaka theorem is that certain average volume strain around an inclusion caused by eigenstrains in the inclusion can be simply expressed without knowing the explicit spatial dependence of the strain (Dunn and Taya, 1993b). In the Mori–Tanaka approach, the concentration tensor of each phase AMT can be expressed with concentrar tion tensor Adil r that is obtained from the dilute approximation model as:

" AMT r

¼

Adil r

N X c1 I þ cr Adil r

#1 ð22Þ

r¼2

where c1 and cr are the volume fraction for phase 1 and r respectively. 3.2.3. Mori–Tanaka approach for multiphase materials Considering a general multiphase composite composed of N phases, the overall electroelastic modulus tensor can be expressed with a multiple inclusion model (Hori and Nemat-Nasser, 1994):

EE ¼ EE1 þ

N X

cr ðEEr  EE1 ÞAMT r

ð23Þ

r¼2

where AMT is the Mori–Tanaka concentration tensor and r can be calculated with Adil by using Eq. (20). For the r three-phase PSF composite shown in Fig. 2, the doubleinclusion model can be developed by using the theorem of Mori–Tanaka approach (Dunn and Ledbetter, 1995; Hori and Nemat-Nasser, 1994): MT EE ¼ EE1 þ c2 ðEE2  EE1 ÞAMT 2 þ c 3 ðEE3  EE1 ÞA3

Fig. 2. Cross-section structure of a representative volume element (RVE) of the PSF composite for micromechanics analysis.

ð21Þ

where I is the 9  9 identity matrix, EE1 and EEr are the electro-elastic tensor of phase 1 and r respectively, and Sr is the constraint tensor for rth phase. The constraint tensor Sr is piezoelectric analog of the Eshelby’s tensor (Eshelby, 1957). This tensor determines the electro-elastic properties of a composite material as a function of the inclusion geometry (Dunn and Taya, 1993b; Odegard, 2004). The complete expression of the constraint tensor Sr for a fibrous inclusion can be found elsewhere (Dunn and Taya, 1993b).

ð19Þ

where Ar is the concentration tensor of rth phase. The concentration tensor Ar has the following properties for the multiphase materials: N X cr Ar ¼ I

3.2.1. Dilute approximation model For the dilute approximation model, it is assumed that the interaction among the reinforced particles/fibers in a matrix-based composite can be ignored. For the multiphase composites shown in Fig. 2, the concentration tensor of each phase r is expressed as:

ð24Þ

After calculating the perturbation strains caused by eigenstrains in the inclusion, the dilute concentration tensors of the double-inclusion model are formulated as the followings (Dunn and Ledbetter, 1995; Lin and Sodano, 2010):

34

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 3. Demonstration of the modified Rule of Mixture for predicting transverse modulus of the three phase PSF composite with two steps by considering inclusion geometry dimensions and shapes with (a) a basic two-phase cell with rectangular geometries using the vertical modeling approach: Step one: (b) a basic two-phase cell with circular geometries using the similar approach, and (c) two-phase model for predicting the transverse modulus for the core fiber – piezoelectric shell domain X2 3 ¼ X2 þ X3 . Step two: (d) a two phase model with a circular inclusion within a square matrix was developed, (e) two-phase model for predicting the effective transverse modulus of the PSF composite X ¼ X1 þ X2 3 .

Adil 3 ¼ I þ DSU2 þ S3 U3

ð25Þ

  c3 c3 Adil DS U2 þ DSU3 2 ¼ I þ S2  c2 c2

ð26Þ

where the constraint tensors S2 and S3 are calculated with the matrix elastic moduli of the reference phase 1 (Dunn and Ledbetter, 1995). Again, the explicit expression of the constraint sensors S2 and S3 can be found elsewhere (Dunn and Taya, 1993b). In Eqs. (25) and (26), the fourth-order tensor Ui can be expressed with the constraint tensors and volume fractions of inclusions and the electro-elastic modulus tensor of each phase (Dunn and Ledbetter, 1995): "  1  #1 c3 c3 U2 ¼  DS þ ðS3 þ F3 Þ S3  DS þ F3  S3  DS þ F2 c2 c2 ð27Þ "

 1  c3 c3 U3 ¼  ðS2 þ F3 Þ þ DS S3  DS þ F2  S3  DS þ F3 c2 c2

#1

ð28Þ

where

F2 ¼ ðEE2  EE1 Þ1 EE1 ; F3 ¼ ðEE3  EE1 Þ1 EE1

and DS ¼ S3  S2

ð29Þ

Thus, the electro-elastic modulus tensor of the active PSF composites can be calculated with Eqs. (22)–(29) by using the dilute approximation model and the Mori–Tanaka approach.

3.2.4. Extended Rule of Mixture In this study, it was found that the Mori–Tanaka approach has some limitation on predicting the effective eff transverse modulus (C eff 11 ¼ C 22 ). This is mainly caused by that Mori–Tanaka approach only considers the volume fraction and excludes the inclusion shape and size effects on the composite properties. As shown in Fig. 2, the core fiber and piezoelectric shell both have circular cross-section in this square element. Hence, the extended Rule of Mixture (Jacquet et al., 2000) was applied to obtain the effective transverse modulus of the PSF composite by considering the size effect of each phase. For a two-phase material, the classical Rule of Mixture for predicting the longitudinal and transverse moduli of a composite is expressed as followings: f m C eff 33 ¼ C 33 c f þ C 33 ð1  c f Þ

1 C eff 11

¼

cf C f11

þ

ð1  cf Þ Cm 11

ð30Þ ð31Þ

T f m where C eff 33 , C 11 , C 11 , C 11 , and cf are the effective composite longitudinal modulus, composite transverse modulus, fiber reinforcement transverse modulus, matrix transverse modulus, and the volume fraction of the fiber reinforcement, respectively. The superscript in the moduli terms denotes the material phase. The extended Rule of Mixture was developed for the composite with layers reinforced with columns (Jacquet et al., 2000). The basic composite cell with the rectangular dimensions a and a0 is shown in Fig. 3(a). The reinforcement bar with sides b and b0 is located in the center of the cell. With the equal length and width, The reinforcement volume fraction and the square root of reinforcement volume fraction can be expressed 0 0 pffiffiffiffi by: cf ¼ bb and cf ¼ ba ¼ ba0 . The transverse modulus C eff 11 aa0

35

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 4. Finite element (FE) modeling of a representative volume element (RVE) of the PSF composites with ABAQUS, (a) three-dimensional FE model with three bonded phases and (b) the cross section view of the FE model of the RVE with the aspect ratio and volume fraction equivalent to 0.4.

Table 1 Electro-mechanical properties of three phases in the PSF composite. Material

C11 (GPa)

C12 (GPa)

C13 (GPa)

C33 (GPa)

C44 (GPa)

C66 (GPa)

j11 (F/m)

j33 (F/m)

e15 (C/m2)

e31 (C/m2)

e33 (C/m2)

Matrix Carbon PZT-5H

3.86 24 151

2.57 9.7 98

2.57 6.7 96

3.86 250 124

2.57 27 23

2.57 11 23.3

2.8 12 1704

2.8 12 1433

0 0 17

0 0 5.1

0 0 27

in the horizontal or vertical direction of this layered composite was calculated with a laminate made of one layer A with pure matrix and one layer B reinforced with the inclusion as (Jacquet et al., 2000):

C eff 11 ¼

Cm 11

þ

C f11 C m 11 pffiffiffiffi pffiffiffiffi f C 11 ð1  cf Þ= cf

þ ð1 

pffiffiffiffi m cf ÞC 11

cf ¼

p p

¼

R2b R2a

and

Rb pffiffiffiffi ¼ cf Ra

C 211

C 311 ð1

R23 R22

p

ð32Þ

The extended Rule of Mixture was applied in this study to formulate the transverse modulus of three-phase PSF composite as shown in Fig. 3. In this study, the prediction of transverse modulus of three-phase PSF composites was conducted with two analytical steps. Firstly, the extended Rule of Mixture was applied to predict the transverse modulus of the two-phase circular composite as shown in Fig. 3(b). A circular cell (with radius Ra ) has a circular reinforcement with the radius Rb , the reinforcement volume fraction is calculated with the uniform thickness:

R2b R2a

C 311 C 211 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ  c3=ð2—3Þ Þ= c3=ð2—3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ ð1  c3=ð2—3Þ ÞC 11

C 2—3 11 ¼

ð33Þ

and the transverse modulus C eff 11 in this circular composite can be obtained with the same format as shown in Eq. (32). The first analytical step used this equation to find the transverse modulus for the domain X2—3 ¼ X2 þ X3 , which is made up of phase 2 and 3. The effective transverse modulus C 2—3 11 for two-phase composite (shown in Fig. 3(c)) can be written as the followings:

ð34Þ

R23 R22

and c3=ð2—3Þ ¼ p ¼ can be calculated with the core fiber and piezo shell dimensions (shown in Fig. 3(c)). Secondly, the extended Rule of Mixture was applied again to predict the transverse modulus of the two-phase square composite with circular inclusion as shown in Fig. 3(d). The square composite (with length b) has a circular reinforcement with the radius Rb , the reinforcement volume fraction is calculated with the uniform thickness:

cf ¼

pR2b 2

b

!

pffiffiffiffi Rb cf ¼

pffiffiffiffi

p

ð35Þ

b

and the transverse modulus C eff 11 in this square composite with a circular inclusion can be obtained with the updated volume fracture in the same format as shown in Eq. (32). The effective transverse modulus for the PSF composite (X ¼ X1 þ X2—3 ) can then be calculated with a square cell with a circular domain X2—3 , as shown in Fig. 3(e). The volume fraction of the domain X2—3 in this square cell was obtained as the following:

c2—3 ¼

pR22 b

2

!

pffiffiffiffiffiffiffiffiffi R2 c2—3 ¼

pffiffiffiffi

p

b

ð36Þ

36

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Table 2 Mechanical displacement and electrical potential boundary conditions applied in the RVE FE model for determining the effective electroelastic properties of PZT-5H composite. Electroelastic properties

Mechanical displacement and electrical potential boundary conditions

Transverse modulus C11

u1(x = 0) = 0 u2(y = 0; y = a) = 0 u3(z = 0; z = c) = 0 u1(x = a) = e110a uðBÞ ¼ 0

Longitudinal modulus C33

u1(x = 0; x = a) = 0 u2(y = 0; y = a) = 0 u3(z = 0) = 0 u1 ðx ¼ aÞ ¼ e033 c uðBÞ ¼ 0

Longitudinal shear modulus C44

u1(x = 0; x = a) = 0 u2 ðz ¼ 0Þ ¼ 0; u2 ðz ¼ cÞ ¼ c:ð12 c023 Þ u3 ðy ¼ 0Þ ¼ 0; u3 ðy ¼ aÞ ¼ a:ð12 c023 Þ

uðBÞ ¼ 0 Transverse shear modulus C66

u1 ðy ¼ 0Þ ¼ 0; u1 ðy ¼ aÞ ¼ a:ð12 c012 Þ u2 ðx ¼ 0Þ ¼ 0; u2 ðx ¼ aÞ ¼ a:ð12 c012 Þ u3 ðz ¼ 0Þ ¼ 0; uðz ¼ cÞ ¼ u0 uðBÞ ¼ 0

Longitudinal dielectric constant j33

u1(x = 0; x = a) = 0 u2(y = 0; y = a) = 0 u3(z = 0; z = c) = 0 u1 ðx ¼ aÞ ¼ e033 c uðz ¼ 0Þ ¼ 0; uðz ¼ cÞ ¼ u0

Piezoelectric straincoupling constant d33

u1(x = 0) = 0 u2(y = 0) = 0 u3(z = 0) = 0 uðz ¼ 0Þ ¼ 0; uðz ¼ cÞ ¼ u0

Thus, the effective transverse modulus of the PSF composite C eff 11 can be predicted with the volume fraction (c2–3) and the calculated modulus C 2—3 for the domain X2—3 as 11 the following,

C eff 11 ¼

C 111

þ

1 C 2—3 11 C 11 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2—3 C 11 ð1  c2—3 Þ= c2—3

þ ð1 

pffiffiffiffiffiffiffiffiffi 1 c2—3 ÞC 11 ð37Þ

C 111

where phase 1.

represents the transverse modulus on matrix

4. Finite element model After discussed the micromechanics model for predicting the electroelastic properties of the three-phase PSF composite in Section 3, the finite element analysis (FEA) was also conducted to simulate the detailed electromechanical behavior with a representative volume element (RVE) model. The micromechanics model can quickly predict the electroelastic properties with theoretical formulations and assumptions. However, the RVE FEA can provide the detailed stress–strain and piezoelectric field distribution for accurate prediction of electroelastic properties (Odegard, 2004). The finite element (FE) model was built with the ABAQUS commercial software (ABAQUS, 2010). The RVE FE models of the PSF composites were built with the ABAQUS software as shown in Fig. 4. One RVE FE model with the as-

pect ratio a = 0.4 and the fiber volume fraction C f = 0.4 is shown in the figure. In the RVE FE model, the global coordinates of x and y represent transverse directions on the cross-section, while z represents the longitudinal direction. All the cubic RVE FE models were constructed with three phases: one cylindrical core fiber, one coated piezoelectric ceramic shell and the surrounding polymer matrix. The material properties selected for each phase are listed in Table 1. The elastic and dielectric properties of carbon fibers, PZT-5H piezoelectric shell and the epoxy polymer matrix were input for FE model simulation as listed in the Table. In addition, the piezoelectric coupling constant vector e of the PZT-5H shell were also applied to analyze electroelastic properties of the active PSF composites. The RVE model was meshed with three-dimensional (3D) 20-noded quadratic brick elements for three phases. In the PZT-5H shell, the piezoelectric elements were used with three displacement degrees of freedom and an electrical degree of freedom. Along the interfaces within three phases of the RVE FE model, the perfect bonding conditions were applied by sharing neighboring element nodes (Lee et al., 2005; Lin and Sodano, 2010). The element mesh was generated for varying model geometries. And the energy or deformation calculated with specific boundary conditions was used to determine each electroelastic property. The FEA was conducted with RVE models of the PSF composites to calculate the electroelastic properties and to compare with micromechanics analysis. In this study, the effective transverse modulus (C 11 ), the effective

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

37

Fig. 5. Displacement contours of the RVE FE model for predicting the properties of (a) effective longitudinal modulus, C 33 , (b) effective transverse modulus, C 11 , (c) effective longitudinal shear modulus, C 44 , (d) transverse shear modulus, C 66 , and (e) effective piezoelectric constant, d33 for the RVE with aspect ratio and fiber volume fraction equivalent to 0.4.

longitudinal modulus (C 33 ), the effective longitudinal shear modulus (C 44 ), the effective transverse shear modulus (C 66 ), the effective dielectric constant (j33 ) and the effec-

tive piezoelectric strain-coupling constant (d33 ) were investigated, as listed in Table 2. In order to study the electroelastic properties varying with geometry factors, the

38

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 5 (continued)

RVE FE models were built with different aspect ratios (a ¼ 0:4; 0:5; 0:6; 0:7) and fiber volume fractions (cf,p = 0.2, 0.4, 0.6, 0.7). The fiber volume fraction is defined as the volume ratio of the active structural fiber (which composed of core fiber and piezoelectric shell or domain X2;3 as shown in Fig. 2) with respect to the RVE. In these RVE FE models, the core fiber dimension was kept as con-

stant as 0.33 mm in radius, while the thickness of piezoletric shell and the side length of the square cross-section were updated for different aspect ratios or volume fraction. For all geometry cases, a 5 mm length in the longitudinal direction was used for the RVE FE models. The energy approach has been applied to determine the effective modulus and dielectric constant of the piezoelec-

39

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 5 (continued)

tric composites (Lin and Sodano, 2010; Odegard, 2004). The strain energy and the electrostatic energy for piezoelectric composites are, respectively,

Ue ¼

V C ijkl eij ekl 2

and U d ¼

V jij Ei Ej 2

ð38Þ

where V is the volume of the RVE, C and j are the effective elastic modulus and dielectric tensor, and e and E are the strain and the electric field tensors applied on the RVE. This approach was also used to determine the effective electroelastic properties of the PSF composites with the RVE FE modeling. The mechanical displacement and electrical potential boundary conditions were applied for each property calculation as listed in Table 2. The RVE FE analysis was conducted to calculate the electroelastic properties by applying the specific boundary conditions (listed in Table 2). Fig. 5 illustrates the deformed RVE FE model with displacement contours for predicting each property. For example, the effective transverse modulus C 11 and the longitudinal modulus C 33 can be calculated with the strain energy by applying the specific boundary conditions

Ue ¼

V C 11 ðe011 Þ2 2

and U e ¼

V C 33 ðe033 Þ2 2

ð39Þ

The longitudinal and transverse displacement contours with value legends and the deformed RVE FE models for computing the effective transverse modulus C 11 and the

longitudinal modulus C 33 are demonstrated in Fig. 5(a) and (b), respectively. Similarly, the effective longitudinal shear modulus C 44 and transverse shear modulus C 66 can also be calculated under the applied boundary conditions as followings:

Ue ¼

V C 44 ðc013 Þ2 2

and U e ¼

V C 66 ðc012 Þ2 2

ð40Þ

Fig. 5(c) and (d) shows the applied displacement magnitude contours with value legends and the deformed FE models for calculating the longitudinal shear modulus C 44 and transverse shear modulus C 66 , respectively. The effective dielectric constant (j33) can be determined via the calculated electrostatic energy, as following:

Ud ¼

u 2 V j33 0 2 d

ð41Þ

where d is the distance between the two surfaces that exposed to the electric field, and u0 is the electric potential applied on the RVE model. The FEA with the ABAQUS software can calculate the total strain energy and electrostatic energy for each RVE model under specific boundary conditions. In order to directly calculate the piezoelectric strain coupling coefficient (dnij ), the FEA was conducted with the applied piezoelectric stress-field material properties of PZT-5H. The effective piezoelectric strain-coupling con-

40

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 6. Comparison of the predicted effective transverse modulus C 11 of the PSF composite between micromechanics analysis and finite element modeling, and the value varying with (a) fiber volume fractions and (b) fiber aspect ratios.

stants dnij can be determined with the strain-electric field relation, expresses as following:

eij ¼ dnij En

ð42Þ

In this study, the effective piezoelectric strain-coupling constant d33 of the composite RVE can be calculated as d33 ¼ E3 =e33 under the applied boundary conditions (Table 2). As shown in Fig. 5(d), longitudinal displacements were generated as the electric field was applied on the normal surfaces of the RVE model. The longitudinal displacement is non-uniform on the free normal surface. The axial strain e33 was determined by averaging the weighted nodal displacements on this surface.

5. Results and discussion In this section, the electro-elastic properties predicted using the micromechanics approach were compared with the RVE FE analysis results. To calculate the effective transverse modulus C 11 , the constant transverse strain was applied in the x-direction. And the y- and z- displacements were fixed for the FE model shown C 11 in Fig. 5(b). The dilute approximate model with Mori–Tanaka approach and the extended Rule of Mixture were used to predict the transverse properties, respectively. The comparison was conducted with the FE predicted modulus C 11 of PSF composites and the micromechanics analysis from these two methods (shown in Fig. 6). Fig. 6(a) shows that the

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

41

Fig. 7. Comparison of the predicted effective longitudinal modulus C 33 of the PSF composite between micromechanics analysis and finite element modeling, and the value varying with (a) fiber volume fractions and (b) fiber aspect ratios.

modulus increases with the fiber volume fraction cf due to less stiffness in the polymer matrix. The comparison results demonstrate that the Mori–Tanaka approach underestimates the modulus C 11 with the average relative difference about 21.9%, with respect to FEA. However, the prediction with the extended Rule of Mixture displays a good comparison with the FEA, with an average relative error of approximately 4.0%. The extended Rule of Mixture has been applied with a two-step procedure to calculate the transverse modulus of the PSF composites. The comparison results indicate that the extended Rule of Mixture can better predict the transverse modulus by including the inclusion size effects. It can be found that the larger difference occurs when the volume fraction are between 0.6 and 0.7 (shown in Fig. 6(a)). Both micromechanics approaches generate less

accurate prediction for the high-volume fiber content. As shown in Fig. 6(b), the effective transverse modulus C 11 increases with the aspect ratio a. This is caused by the fact: matrix C piezo > C core . This study shows the similar modu11 > C 11 11 lus-aspect ratio trend as other work such as (Lin and Sodano, 2010). The effective longitudinal modulus C 33 was predicted by applying constant axial strain and fixing displacements in other directions in the FE model shown in Fig. 5(a). Only the dilute approximate model with Mori–Tanaka approach was applied to calculate the properties for a range of fiber volume fractions and aspect ratios. The FEA results were compared with the micromechanics analysis and the Mori–Tanaka approach (as shown in Fig. 7). The average relative difference is about 0.6%. The comparison results indicated the Mori–Tanaka approach can achieve accurate

42

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 8. Comparison of the predicted effective shear modulus C 44 of the PSF composite between micromechanics analysis and finite element modeling, and the value varying with (a) fiber volume fractions and (b) fiber aspect ratios.

prediction of the longitudinal modulus C 33 with only limited relative error for fiber volume fraction between 0.6 and 0.7. The modulus C 33 increases with the fiber volume fraction due to lower matrix longitudinal stiffness as shown in Fig. 7(a). The modulus C 33 decreases with the aspect rations because the piezoelectric materials have lower longitudinal stiffness compared to the core carbon fiber. To calculate the effective longitudinal shear modulus C 44 , a constant shear strain e023 was applied and the x-displacements were fixed in the RVE FE model. The comparison of the predicted effective modulus C 44 of the PSF composite with micromechanics analysis and FE modeling was shown in Fig. 8. Fig. 8(a) shows the prediction with the dilute approximate model with Mori–Tanaka approach has good agreement with the FE analysis. The average relative difference is about 2.9% when the fiber volume fraction is less than 0.6. However, the average relative difference

reaches 8.0% as the fiber volume fraction is about 0.7. The comparison results indicate that the Mori–Tanaka approach is less accurate when larger fiber volume is used in the RVE model. By checking Fig. 8(b), the difference increases with the fiber volume fraction and aspect ratio. The relative difference trends are almost identical for different fiber aspect ratio cases because the shear modulus C 44 for PZT-5H and carbon core fiber are very close as shown in Table 1. The effective transverse shear modulus C 66 was calculated with the applied constant strain e012 and the fixed zdisplacement in the RVE model. The comparison of the predicted effective modulus C 66 of the PSF composite with micromechanics analysis and FE modeling was shown in Fig. 9. The modulus C 66 increases with the fiber volume fraction and aspect ratio due to the material properties as piezo C core > C matrix . Fig. 9 shows that the dilute approxi66 > C 66 66

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

43

Fig. 9. Comparison of the predicted effective shear modulus C 66 of the PSF composite between micromechanics analysis and finite element modeling, and the value varying with (a) fiber volume fractions and (b) fiber aspect ratios.

mate model with Mori–Tanaka approach overestimates the effective transverse shear modulus C 66 by comparing to the FEA results for all fiber volume fraction and aspect ratio cases. As shown in Fig. 9(a) and (b), the average relative difference increases from 3.1% to 10.1% with the fiber volume fraction (up to 0.6) and then decreases to about 3.6% at a volume fraction of 0.7. This is contributed by that the strain energy rapidly increases at a fiber volume fraction about 0.7 in the FEA. In contrast to the previous case, the effective transverse shear modulus increases as the aspect ratio, because the piezoelectric material has higher transverse shear modulus than the core fiber. In the RVE FE model, the effective longitudinal dielectric constant j33 was calculated with the electrostatic energy by applying a constant electric field along the z-direction and fixing all the displacements. A zero potential and a po-

sitive potential were applied on the surface (z ¼ 0) and the opposite surface (z ¼ c) in the longitudinal direction, respectively. In this study, the relative dielectric constant was evaluated as j33 =j0 , where j0 is the permittivity constant equivalents to 8:85  1012 F=m. Fig. 10 shows a strong agreement between the dilute approximate model with Mori–Tanaka approach and the FE model for all volume fractions and aspect ratios, with an average relative difference about 1.9%. It can be noticed that the Mori–Tanaka model slightly overestimates the dielectric properties. The dielectric property increases as the volume fraction and aspect ratio, because the piezoelectric material has the highest dielectric constant among the three phases of the PSF composites. In the Section 4 of the FE analysis, the piezoelectric field-stress properties were defined for the PZT-5H mate-

44

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Fig. 10. Comparison of the predicted effective dielectric constant j33 of the PSF composite with micromechanics analysis and finite element modeling, and the value varying with (a) fiber volume fractions and (b) fiber aspect ratios.

rial. In the FE model, an electrical potential was applied on the normal surface (z = c) and the displacements and the electrical potential were fixed on the opposite normal surface (z = 0) in the RVE FE model. Following the strain-electric field relation in Eq. (42), the effective piezoelectric lam strain coupling constant d33 can be calculated with the weighted average normal strain ezz on the normal surface (z = c). The piezoelectric strain coupling constant d33 of the PZT-5H material can be determined as 619 pC/m with the relation (15). The effective piezoelectric coupling ratio lam d33 =d33 of the PSF composites were compared with micromechanics analysis and the FE modeling as shown in Fig. 11. The prediction with the dilute approximate model with Mori–Tanaka approach has good agreement for the aspect ratios of 0.4, 0.5 and 0.6 with different volume fractions. The average relative difference between

Mori–Tanake approach and FEA is about 2.6% for these aspect ratios. For the case of the aspect ratio of 0.7, the Mori– Tanaka approach overestimates the piezoelectric coupling lam constant d33 . The average relative difference is about 5.6%. The comparison results again indicate that the dilute approximate model with Mori–Tanaka approach has less prediction with larger fiber volume used in the RVE model. The figure also shows that the piezoelectric coupling ratio increases with the fiber aspect ratio and the volume of active piezoelectric component. The above compassion results indicate that the dilute approximate model with Mori–Tanaka approach can generate close prediction on effective electro-elastic moduli by comparing the FEA. However, the Mori–Tanaka approach has some limitation on predicting the effective transverse modulus of the multiphase PSF composites.

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

45

lam

Fig. 11. Comparison of the predicted effective piezoelectric coupling ratio d33 =d33 of the PSF composite from the micromechanics analysis and finite element modeling.

The prediction on the transverse modulus was improved by employing the Extended Rule of Mixture with a twostep procedure in this study. 6. Conclusions To overcome the fragile nature of the monolithic piezoelectric material, the PSFs were fabricated by coating the piezoceramic onto carbon/SiC core fibers. The piezoelectric structural fibers were deployed unidirectionally into the polymer matrix. Due to inhomogeneous property of the PSF composite, it is essential to predict the effective moduli of the composite material properties. In this study, the electro-elastic properties including effective moduli, dielectric constants and piezoelectric coupling factors of the PSF composite were predicted with micromechanics analysis and finite element simulation of the RVE model. The micromechanics analytical model was developed with the combination of dilute approximation model, Mori–Tanaka approach, and extended Rule of Mixture. The dilute approximate model with Mori–Tanaka approach can efficiently predict the electroelastic properties except the transverse modulus of the PSF composites. The twostep analysis procedure with the extended Rule of Mixture was formulated to accurately predict the transverse modulus by considering the inclusion size effects. The FE simulation of the RVE model was conducted to represent the actual structure and component properties of the PSF composites with the commercial ABAQUS software. The energy approach method was applied to predict electroelastic properties including the effective moduli and the dielectric constants of the PSF composite. The effective piezoelectric strain coupling parameter was calculated with the strainelectric field relation with the RVE FE modeling. The prediction on the effective longitudinal modulus C 33 , the effective longitudinal shear modulus C 44 and the

effective transverse shear modulus C 66 from the dilute approximate model with Mori–Tanaka approach has good comparison from the FE modeling. As shown in Figs. 7–9, the average relative difference between micromechanics analysis and the FEA are calculated as 0.6%, 2.9%, and 6.2% for C 33 , C 44 and C 66 respectively. However, the extended Rule of Mixtures considering the effects of the inclusion size and volume fraction has closer prediction on the effective transverse modulus C 11 than the Mori–Tanaka approach, with comparison of the FE modeling results. The prediction of the relative dielectric constant j33 =j0 from the dilute approximate model with Mori–Tanaka approach was favorably compared with the energy-based FEA, with the average relative difference about 1.9%. The FE analysis on the effective piezoelectric strain coupling constant ratio lam of d33 =d33 with strain-electric field relation in Eq. (42), was also close to the prediction from the dilute approximate model with Mori–Tanaka approach. Overall, the combined micromechanics model can favorably predict the effective electro-elastic properties of the PSF composite. Therefore, the proposed combined micromechanics model and the finite element analysis can be applied to study the constitutive properties of the multiphase PSF composites. Acknowledgement The authors would like to acknowledge the partial financial support from the Michigan Space Grant. References ABAQUS, 2010. Hibitt, Karlsson and Sorenson. Version 6.9, Pawtucket, R.I. Bent, A.A., 1994. Piezoelectric fiber composites for structural actuation, Dept. of Aeronautics and Astronautics. Massachusetts Institute of Technology, p. 204. Bent, A.A., 1997. Active fiber composites for structural actuation. Massachusetts Institute of Technology, Ph.D., p. 209.

46

Q. Dai, K. Ng / Mechanics of Materials 53 (2012) 29–46

Bent, A.A., Hagood, N.W., 1997. Piezoelectric fiber composites with interdigitated electrodes. Journal of Intelligent Material Systems and Structures 8, 903–919. Brei, D., Cannon, B.J., 2004. Piezoceramic hollow fiber active composites. Composites Science and Technology 64, 245–261. Dai, Q., Ng, K., 2010. Micromechanical Analysis of Damping Performance of Piezoelectric Structural Fiber Composites, 2010 SPIE Smart Structure/NDE Conference, San Diego, CA. Dinzart, F., Sabar, H., 2009. Electroelastic behavior of piezoelectric composites with coated reinforcements: micromechanical approach and applications. International Journal of Solids and Structures 46, 3556–3564. Dunn, M.L., Ledbetter, H., 1995. Elastic moduli of composites reinforced by multiphase particles. Journal of Applied Mechanics 62, 1023–1028. Dunn, M.L., Taya, M., 1993a. An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences 443, 265–287. Dunn, M.L., Taya, M., 1993b. Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures 30, 161–175. Dunn, M.L., Wienecke, H.A., 1996. Green’s functions for transversely isotropic piezoelectric solids. International Journal of Solids and Structures 33, 4571–4581. Dunn, M.L., Wienecke, H.A., 1997. Inclusions and inhomogeneities in transversely isotropic piezoelectric solids. International Journal of Solids and Structures 34, 3571–3582. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London Series A. Mathematical and Physical Sciences 241, 376–396. Fakri, N., Azrar, L., El Bakkali, L., 2003. Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements. International Journal of Solids and Structures 40, 361–384. Hori, M., Nemat-Nasser, S., 1994. Double-inclusion model and overall moduli of multi-phase composites. Journal of Engineering Materials and Technology 116, 305–309. Huang, J.H., Kuo, W.-S., 1996. Micromechanics determination of the effective properties of piezoelectric composites containing spatially oriented short fibers. Acta Materialia 44, 4889–4898.

IEEE, 1988. IEEE Standard on Piezoelectricity, ANSI/IEEE Std 176–1987, p. 0–1. Jacquet, E., Trivaudey, F., Varchon, D., 2000. Calculation of the transverse modulus of a unidirectional composite material and of the modulus of an aggregate. Application of the rule of mixtures. Composites Science and Technology 60, 345–350. Kar-Gupta, R., Venkatesh, T.A., 2007. Electromechanical response of 1–3 piezoelectric composites: an analytical model. Acta Materialia 55, 1093–1108. Lee, J., Boyd Iv, J.G., Lagoudas, D.C., 2005. Effective properties of threephase electro-magneto-elastic composites. International Journal of Engineering Science 43, 790–825. Li, J.Y., 1999. On micromechanics approximation for the effective thermoelastic moduli of multi-phase composite materials. Mechanics of Materials 31, 149–159. Li, J.Y., 2000. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science 38, 1993–2011. Lin, Y., Sodano, H.A., 2008. Concept and model of a piezoelectric structural fiber for multifunctional composites. Composites Science and Technology 68, 1911–1918. Lin, Y., Sodano, H.A., 2009. Electromechanical characterization of a active structural fiber lamina for multifunctional composites. Composites Science and Technology 69, 1825–1830. Lin, Y., Sodano, H.A., 2010. A double inclusion model for multiphase piezoelectric composites. Smart Material Structures, 19. Lu, H.A., Wills, L.A., Wessels, B.W., Lin, W.P., Zhang, T.G., Wong, G.K., Neumayer, D.A., Marks, T.J., 1993. Second harmonic generation of poled BaTiO3 thin films. Applied Physics Letters 62, 1314–1316. Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21, 571–574. Odegard, G.M., 2004. Constitutive modeling of piezoelectric polymer composites. Acta Materialia 52, 5315–5330. Reza Moheimani, S.O., Fleming, A.J., 2006. Piezoelectric Transducers for Vibration Control and Damping. Spinger. Takagi, K., Sato, H., Saigo, M., 2001. Robust vibration control of the metalcore-assisted piezoelectric fiber embedded in CFRP composite in: Smith, R.C. (Ed.), Smart Structures and Materials 2004: Modeling, Signal Processing, and, Control, pp. 376–385.