Multi-Inclusion modeling of multiphase piezoelectric composites

Composites: Part B 47 (2013) 181–189

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Multi-Inclusion modeling of multiphase piezoelectric composites Mohammad H. Malakooti a,⇑, Henry A. Sodano a,b a b

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250, USA Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611-6250, USA

a r t i c l e

i n f o

Article history: Received 7 June 2012 Received in revised form 13 August 2012 Accepted 7 October 2012 Available online 24 November 2012 Keywords: A. Smart materials C. Micro-mechanics B. Interface/interphase C. Finite element analysis (FEA) A. Nano-structures

a b s t r a c t Recent work on multifunctional materials has shown that a functionally graded interface between the fiber and matrix of a composite material can lead to improved strength and stiffness while simultaneously affording piezoelectric properties to the composite. However the modeling of this functional gradient is difficult through micromechanics models without discretizing the gradient into numerous layers of varying properties. In order to facilitate the design of these multiphase piezoelectric composites, accurate models are required. In this work, Multi-Inclusion models are extended to predict the effective electroelastic properties of multiphase piezoelectric composites. To evaluate the micromechanics modeling results, a three dimensional finite element model of a four-phase piezoelectric composite was created in the commercial finite element software ABAQUS with different volume fractions and aspect ratios. The simulations showed excellent agreement for multiphase piezoelectric composites, and thus the modeling approach has been applied to study the overall electroelastic properties of a composite with zinc oxide nanowires grown on carbon fibers embedded in the polymer. The results of this case study demonstrate the importance of the approach and show the system cannot be accurately modeled with a homogenized interphase. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The desire to reduce the weight and complexity of systems has led to the invention of multifunctional materials which seek to combine the performance objectives of several components into a single material. One class of multifunctional materials is created through the combination of structural composites and piezoelectric materials such that embedded energy generation and storage, vibration and shape control and sensing can be achieved. With piezoelectricity forming the basis for many sensors/actuators, microelectro-mechanical systems, power generators, capacitors and structural health monitoring approaches, this class of materials offers a diverse range of applications [1–3]. The potential for multifunctional materials based on piezoelectricity has been further advanced by the recent demonstration that a functionally graded interface formed through the growth of zinc oxide (ZnO) nanowires on the surface of carbon fiber can lead to a 3.28 times increase in the interface strength [4]. With ZnO being piezoelectric, this demonstration of enhanced strength can offer a shift in the design of multifunctional materials, which oftentimes come at the expense of lost performance in each role. The composite exhibited increased interfacial strength due to reduced stress concentration arising from the graded interface as well as the chemical interac⇑ Corresponding author. E-mail address: malakooti@ufl.edu (M.H. Malakooti). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.10.034

tion between the fiber and ceramic. While the functionally graded interface offers potential advantages, it complicates the identification of the material’s constitutive properties. Here micromechanics models will be developed to predict the coupled electroelastic properties on a composite with a functionally graded piezoceramic interphase surrounding the fiber. Over the last 50 years, micromechanics models for heterogeneous materials have been developed with the purpose of estimation of the effective elastic properties of inhomogeneous materials. Over the last decade, micromechanics models have been extended to the prediction of electrical properties as well as coupling coefficients. Historically, Dunn and Taya [5] have been pioneers in the prediction of effective electroelastic properties of piezoelectric composites with different geometries. They extended the dilute, self-consistent and Mori–Tanaka methods to predict electroelastic properties of piezoelectric composites [5,6]. Hence, coupled fields such as electroelastic, thermoelectric and thermoelastic can be modeled with micromechanics approaches. Moreover, Chen [7] presented a formulation to estimate the overall thermo-electro-elastic moduli of multiphase fibrous composites with the self-consistent and Mori–Tanaka methods, Which introduce the ability of micromechanics to model coupled multi-fields such as thermo-electro-elastic, magneto-electro-elastic, and magneto-thermo-elastic [8,9]. The most commonly predicted bulk property of multiphase composites is the elastic properties of a double-inclusion which is an

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isolated ellipsoidal inclusion containing another inclusion [10]. The formulation was generalized to predict the elastic modulus of Multi-Inclusions and multiphase particles [10,11]. The double inclusion model estimates the effective elastic moduli of three-phase composites accurately [12]. This model has been used for the study other bulk properties such as magnetostriction moduli [13]. However, there are a limited number of studies on the MultiInclusion problems and none on four or more phases of materials. Some of the literature presents generalized formulations and extensions of the double inclusion model to predict coupling coefficients in magneto-electric and thermo-elastic fields, but the case studies and numerical analysis are limited to three-phase composites [14,15]. To fully understand the overall properties of Multi-Inclusion composites including more than three phases, the average field quantities can be obtained by applying the generalized form of the Eshelby [16] solution or Mori–Tanaka results [11,17]. Still there is doubt on the predicted result of the Mori– Tanaka model since it is asserted that the Mori–Tanaka and selfconsistent approaches yield a non-symmetric stiffness matrix, especially in the case of oriented fibers embedded in the matrix [14,18]. The mathematical derivation of the Mori–Tanaka and Multi-Inclusion models is complex [19], but the predicted results show agreement with numerical and experimental results [20,21]. Recently, Lin and Sodano [21,22] developed a smart material termed the active structural fiber (ASF) to combine the advantages of composite reinforcement and the sensing and actuation properties of piezoelectric materials. Later, the authors developed a one dimensional micromechanics model to predict the effective longitudinal piezoelectric coupling of the ASF [21] and the results were validated using both finite element analysis (FEA) and experimental approaches. They have also studied the three dimensional electroelastic properties of the piezoelectric composite with three different phases. Hence, the electroelastic moduli of active structural fibers were predicted accurately through a double inclusion model [21]. Later, they demonstrated that the growth of vertically aligned piezoelectric nanowires on the surface of a fiber could lead to up to 3.28 times increased interfacial strength providing not only piezoelectric functionality but also improved strength [4]. The modeling of a fiber coated in nanowires can be approximated as a functional gradient when viewed as a composite of varying volume fraction of the reinforcing and matrix phases. For this class of interface it is unclear if the existing micromechanics models can effectively predict the constitutive properties of a fiber surrounded by a non-homogenous inclusion. In this paper, the electroelastic properties of a multiphase composite are determined through two micromechanics models, and validation of models is done by finite element analysis for various coating thickness. The finite element modeling procedure is presented in detail to evaluate the predicted results by micromechanics. Since the Multi-Inclusion models show good agreement in prediction of the electroelastic properties with finite elements, the interface of the ZnO nanowires grown on the carbon fiber and embedded in a polymer is modeled with a large number of layers. The required number of discretized layers in order to predict the overall electroelastic properties accurately and also the effect of the length of ZnO nanowires on the discretization are investigated. 2. Piezoelectric constitutive equations Although there are four separate forms of the linear piezoelectric constitutive equations the mechanical stress and electric displacement tensors explicitly, are commonly defined as:

rij ¼ C ijmn emn  enij En Di ¼ eimn emn þ jin En

where rij , emn , En and Di are the stress tensor, strain tensor, electric field vector and electric displacement vector, respectively. Cijmn, eimn and jin are elastic (measured in a constant electric field), piezoelectric tensor (measured at a constant strain or electric field) and the dielectric moduli (measured at a constant strain), respectively. It is more convenient to combine electrical and mechanical properties into a single matrix for micromechanics modeling of heterogeneous electronic composites. Considering the orthotropic nature of different components of piezoelectric composites (polymer, piezoelectric and carbon fiber in this case) the general constitutive equation in matrix format can be expressed as:

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 2

0 0

0 0

0 0

0 0

C 23 C 33 0

0

0

0

0

0

0

e15

0

0

C 44 0

0

0

0

C 55 0

0

0

0

0

0

0

0

e15 0

0

0

e15 0

e31 e32 e33 0

D3

0

e15 0

C 66 0

0

j1

0

0

0

j2

0

0

0

32 3 e31 e11 6e 7 e32 7 76 22 7 76 7 6 7 e33 7 76 e33 7 76 7 0 76 c23 7 76 7 76 c 7 0 76 13 7 76 7 0 76 c12 7 76 7 76 E1 7 0 76 7 76 7 0 54 E2 5 j3 E3

ð3Þ

To solve the piezoelectric inhomogeneity problems, the elastic and electric variables can be treated as a unique field variable [23]. Thus Eq. (3) is simplified to R = EZ, where R is a tensor including stress and electric displacement fields and Z is elastic strain and electric field. Boldface E is the electroelastic property matrix, and can be expressed as:

E¼



C 66 e36

et63



j33

ð4Þ

where C, e and j are the elastic stiffness, piezoelectric constant, and relative permittivity matrices, respectively. et represents the transpose of the piezoelectric matrix. The detailed electroelastic constitutive equations of piezoelectric materials can be found elsewhere [5]. 3. Micromechanics modeling The micromechanics approach is considered for heterogeneous materials with more than two phases while the design of multiphase composites with customized properties requires a mathematical model with multi components. In terms of validity and the cost of numerical calculations, two micromechanics models, Mori–Tanaka (MT) and Multi-Inclusion (MI), are preferred over other methods such as the self-consistent, dilute, and differential techniques. The formulations of both methods are extended for coupled behavior in the material and presented in the most convenient formulation for use in modeling any coupled behavior of multiphase materials. 3.1. Extended Mori-Tanka model for coupled behavior The assumption of the Mori–Tanka model is that the average strain in each component is equal to the single component embedded in the infinite volume which is subjected to a uniform strain. If a heterogeneous material with n phases is assumed, the volume averaged fields in the composite can be defined as:

¼ R

n X r fr R

Z ¼

n X fr Z r

r¼1

ð1Þ ð2Þ

C 12 C 13 0 C 22 C 23 0

r¼1

ð5Þ

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Next, the piezoelectric constitutive equation is expressed for averaged fields and displacements over representative volume element  ¼ EZ,  by defining the concentration tensor (Ar) for each (RVE) as R phase of the composite, the volume–average strain and electric fields of phase r is related to the overall fields of the RVE by the following equation:

Z r ¼ Ar Z

ð6Þ

where Ar is the concentration matrix with the following property: n X f r Ar ¼ I

ð7Þ

r¼1

Finally, the Mori–Tanaka model which is able to determine the overall electroelastic properties in terms of volume fractions (fr), concentration tensor (Ar), and electroelastic modulus of each component is expressed as:

E ¼ E1 þ

n X

fr ðEr  E1 ÞAr

ð8Þ

3.2. Extended Multi-Inclusion model for coupled behavior The Multi-Inclusion model is the generalized form of the double inclusion model to estimate the elastic behavior of composite inclusions. In this paper, the MI model will be applied to estimate the electroelastic properties of piezoelectric composites with four annular piezoelectric layers. A Multi-Inclusion is an inclusion embedded in number of different inclusions where each layer can have arbitrary electroelastic properties and shape. Assuming an ellipsoidal geometry for the cross section of the Multi-Inclusion, the internal inclusions are defined by a series of ellipsoids termed as Xi(i = 1, 2, . . ., n). To clarify this definition an inclusion with four phases has been shown in Fig. 1. The inclusion is working in an infinite domain of uniform electrical and mechanical field. For a piezoelectric/polymer composite, the infinite domain is the polymer matrix which is isotropic with all electrical properties being zero. The electroelastic tensor is formulated as:

(

r¼2

AMT r Þ

The Mori–Tanaka concentration tensor ðAr ¼ is defined as a function of dilute concentration tensor ðAdil r Þ. The mathematical expression of the basic assumption (constant strain field) in the Mori–Tanaka approach shows up in the dilute concentration tensor in term of new tensor termed the Eshelby tensor:

" AMT r

¼

Adil r

n X f1 I þ fr Adil r

#1 ð9Þ

r¼2

h i1 1 1 r 1 Adil r ¼ I þ Sr ðE Þ ðE  E Þ

ð10Þ

E¼

Ei E

1

x in Ci ði ¼ 2; 3; . . . ; nÞ

ð12Þ

surrounded domain

where Ci is the region with different electroelastic properties. To compute the effective properties of multiphase composites, it is assumed that bonding is perfect between phases and the eigenstrain and electrical potential fields are constant. Each region participates in the overall properties with its volume fraction defined as fi=Ci /Vtotal and bulk properties. The overall electroelastic properties for a n phase composite can be estimated by applying Multi-Inclusion model:

( EMI ¼ E0 :

Ið9Þ þ

n X fi ðS  Ið9Þ ÞðAi  SÞ1 i¼2

)

where Sr is the Eshelby tensor of phase r. The Eshelby tensor is used to determine the elastic and electrical properties of composites considering the geometry of the inclusion in a matrix. Several authors have extended Eshelby’s classical solution for single inclusion composites to piezoelectric inclusions [24,25]. The Eshelby tensor for ellipsoid inclusions within an isotropic matrix is a function of ellipsoid axes ðari Þ of phase r as well as elastic properties of the infinite surrounded domain (E1), as:

where EMI is the estimated property matrix presented in Eq. (3), S is the Eshelby tensor, I(9) is the nine by nine identity matrix, E1 is reserved for the electroelastic matrix of the infinite domain and Ai can be defined as:

Sr ¼ FðE1 ; ar1 ; ar2 ; ar3 Þ

Ai ¼ ðE1  Ei Þ1 i ¼ 2; 3; . . . ; n

ð11Þ

For this study, the analysis is restricted to the transversely isotropic materials with circular cross sections, see Fig. 1a. For elliptical long inclusions, the Eshelby tensor is a function of E1 and elliptic aspect ratio (a1/a2). However, for circular cross sections this aspect ratio is one, so for orthotropic materials the Eshelby tensor is only a function of the matrix properties.

( :

I

ð9Þ

þ

n X

)1 ð9Þ

i

fi ðS  I ÞðA  SÞ

1

ð13Þ

i¼2

ð14Þ

This model has the same formulation for a composite consisting of cylindrical aligned inclusions and multiphase particle composites with the geometrical differences appearing in the Eshelby tensor. Detailed formulation of the Multi-Inclusion model can be found elsewhere [17]. The effective electroelastic moduli of fourphase piezoelectric/polymer composites are presented based on

Fig. 1. (a) Three-phase cylindrical inclusion with specified direction. (b) Four-phase inclusion embedded in an infinite matrix with different electroelastic properties.

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the MT and MI model in the simulation results section. The advantage of the MI model is that no limitation is placed on the number of inclusions, thus this model can be used for modeling the numerous discretized layers of the graded interfaces. 4. Finite element modeling The accuracy of the micromechanics models can be evaluated through finite element analysis. Since FEA is capable of determining the energy and deformations of the RVE numerically, it can be used to estimate the effective electromechanical properties. FEA predicts the overall electroelastic moduli of the RVE and the stress and strain field of piezoelectric inclusions accurately [20,26,27]. It has also been shown that the overall properties predicted by FEA under the correct boundary conditions agree well with experimental results [28,29]. Thus, the FEA results are assumed to be an accurate reference for validation of the accuracy of the estimated properties by the MI and MT models. Numerical simulations were performed by creating piezoelectric/polymer RVEs in the commercial finite element software ABAQUS. To estimate the effective properties of the RVE, a parametric three dimensional model including four different phases was created (can be seen in Fig. 2). The electroelastic properties of each phase are presented in Table 1 and the volume fraction of the fiber was varied through modification of the matrix size while holding the fiber geometry constant. The matrix is meshed with 20-node brick elements, where each node has three displacement degrees of freedom and one electrical degree of freedom. The assigned material orientations are assumed to be the same as the transversely isotropic inclusion directions as it was shown in Fig. 1. The FEA model returns accurate overall properties for square RVEs under appropriate boundary conditions [9,30–32], hence a three dimensional RVE with square cross section is chosen for study. The RVE strain energy can be acquired directly from ABAQUS under any desired applied loads and displacements. Thus, by controlling the boundary conditions, it is possible to obtain overall mechanical properties of the RVE through a simple strain energy equation. The strain energy is a function of geometry, property and applied strains:

Us ¼

V C ijkl eij ekl 2

ð15Þ

where V is the total volume, Cijkl is effective elastic properties, and both eij and ekl are the applied strains. The proper boundary

Fig. 2. FEA model of multifunctional composite with four different phases.

conditions which have been applied to the RVE in order to determine all electroelastic properties are presented in Table 2. In this case by substituting the values of the produced strain energy Us, RVE volume and applied strain, the Young’s modulus and shear modulus are calculated. Furthermore the symmetry of the RVE reduces the number of independent unknowns of mechanical properties to two values; longitudinal (Y3) and transverse (Y1 = Y2) Young’s modulus. The longitudinal and transverse shear modulus are expressed as G13 = G23 and G12, respectively. Effective dielectric properties can be estimated by applying an electric potential to two sides of the RVE and outputting the stored electrostatic energy from ABAQUS. In this condition the inclusion acts like a capacitor allowing the following formulation to be used to calculate the stored energy by:

Ue ¼

1 A jeff ðu2  u1 Þ2 2 l

ð16Þ

where jeff represents the effective dielectric property of the RVE, A is the area of the faces of the capacitor in common and l is the distance between two faces where electric potential (/1 and /2) is applied to them. Piezoelectric materials are able to convert applied mechanical strain to electrical energy and hence the overall piezoelectric moduli of the RVE can be estimated by applying an electric potential in an appropriate direction and measuring the induced strain:

eij ¼ dnij En

ð17Þ

where En is the applied electric field, eij is the measured strain and dnij is the piezoelectric strain coupling coefficient. The strain in z-direction was estimated as the average of the strain at each node on the surface of the RVE. 5. Simulation results In this section the predicted electroelastic properties of the RVE by two micromechanics models (MI and MT) and FE model are presented. Three different aspect ratios were considered (0.33, 0.50 and 0.66) where the aspect ratio (termed a) is defined as the ratio of the total thickness of all piezoelectric shell layers to the radius of the entire concentric ASF. The effective electroelastic properties of the multifunctional composite with respect to ASF volume fraction are presented in Figs. 3 and 4. The ASF volume fraction is the ratio of the ASF volume (piezoelectric and carbon fiber) to the total volume and is different from the volume fraction of each phase used for the micromechanics formulation. The maximum volume fraction for the square RVE is 78% since a simple volume calculation of an embedded circle in a square reveals that the ASF volume fraction cannot exceed 78% (p/4). To avoid creating an ultra-thin layer of polymer on the close region of the ASF to the square sides, the ASF volume fraction is limited to a maximum of 70%. Thus the MT and MI models in this case are not able to be validated for unitary (100%) reinforcement by FEA results. The longitudinal Young’s modulus Y3 shows good agreement with FEA results for all three aspect ratios and as seen in Fig. 3a linear variation of Y3 with ASF volume fraction is consistent with simple rule of mixture while the transverse Young’s modulus is an exponential function of the ASF volume fraction. Thus, the piezoelectric composite with a larger volume of piezoelectric material has higher transverse modulus. The finite element model is a discretized model with a limited number of elements so the predicted transverse Young’s modulus Y1 = Y2 is slightly higher than the micromechanics model results in Fig. 3b. Increasing the aspect ratio causes higher difference between FEA and model transverse Young’s modulus since the inclusion with higher modulus inside the matrix is more rigid and close to the plane of applied load.

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M.H. Malakooti, H.A. Sodano / Composites: Part B 47 (2013) 181–189 Table 1 Electromechanical properties of materials in the triple inclusion model. Material

C11 (GPa)

C12 (GPa)

C13 (GPa)

C33 (GPa)

C44 (GPa)

C66 (GPa)

j11

j33

e33 (C/m2)

e31 (C/m2)

e15 (C/m2)

Matrix Carbon fiber PZT-5A PZT-7A ZnO

8.1 24 120 158 20.7

5.4 9.7 75 88 11.77

5.4 6.7 75 81 10.61

8.1 243.7 111 125 20.95

1.4 27 21 29.5 4.48

1.4 11 22 35 4.46

2.8 12 1730 930 8.9

2.8 12 1700 425 10.2

0 0 15.8 9.5 0.96

0 0 5.3 2.2 0.62

0 0 12.3 10.6 0.37

Table 2 Applied boundary conditions to the RVE. Property

Displacements

Electric potential

Y1

u1 = e0x1 u2 = u3 = 0

/=0

Y3

u3 = e0x3 u1 = u2 = 0

/=0

G23

u2 = (c0/2)x3 u3 = (c0/2)x2 u1 = 0

/=0

G12

u1 = (c0/2)x2 u2 = (c0/2)x1 u3 = 0

/=0

j11 j33

u(x = 0) = 0 u(z = 0) = 0

/(x = a) = /0 /(z = c) = /0

d31 d33

u(x = 0) = 0 u(z = 0) = 0

/ = /0 z / = /0 z

The trend of the effective transverse shear modulus of RVE (Fig. 3c) implies that the FEA predicted results are almost the same for different aspect ratios since the applied boundary conditions in this condition act only on planes one and two (only the matrix phase). However, the shear modulus in the third direction is a function of the aspect ratio due to the direct collaboration of the piezoelectric phase in the applied shear load as a boundary condition. In general, the predicted shear modulus is not highly sensitive to the thickness of the interface. The functionality of the ASF is due to the piezoelectric phase of the composite materials since these materials have relatively high dielectric constant and coupling coefficient while the flexibility and the strength of the ASF is because of the polymer matrix and carbon fiber respectively. To understand the appropriate volume fraction of each phase to the purpose of new material design with demanded electrical and coupled properties, the predicted dielectric constant and piezoelectric moduli of the RVE are visualized in Fig. 4. The predicted transverse permittivity (Fig. 4a) shows an exponential increase with respect to the ASF volume fraction while the longitudinal permittivity (Fig. 4b) has a linear relationship. The FEA results for the relative permittivity perfectly match with the models of the effective longitudinal permittivity for all three aspect ratios and over the entire range of ASF volume fractions. However, the error between the FEA and micromechanics modeling results in transverse permittivity increases with respect to volume fraction. One of the most important parameters of piezoelectric materials is the electromechanical coupling coefficient in the longitudinal direction d33, which is plotted in Fig. 4c and describes the effect of double piezoelectric phases in the ASF as a function of volume fraction. It is worth mentioning that for evaluating the effective d33 of the RVE, the electrical potential is applied in the three direction and the induced strain in the same direction was determined by averaging the displacement over the entire free surface. Fig. 4c shows excellent agreement not only between FEA and micromechanics modeling results but also the trend in the effective piezoelectric moduli of a four-phase RVE is consistent with the experimental results for low volume fraction of two and threephase composites [33–35].

In this case study, the predicted electroelastic properties by the Mori–Tanaka and Multi-Inclusion models show excellent agreement, however there is a small amount of numerical errors due to the inversion of the matrices, so in Figs. 3 and 4 only one set represents both MI and MT micromechanics models. Hu and Weng [36] showed that for the double-inclusion composite which has the same shape and orientation of enclosed inclusions, the predicted elastic properties by Mori–Tanaka and double-inclusion is the same. Here, in the cylindrical inclusions embedded in a square polymer matrix has the same orientation, thus it can be concluded that the Mori–Tanaka and Multi-Inclusion models prediction for multiphase composites with the same shape and orientation inclusions is the same. To ensure this hypothesis, the analysis of micromechanics models with five and six different cylindrical layers confirmed that the Mori–Tanaka and Multi-Inclusion models equally estimate the overall electroelastic properties of the piezoelectric composites with four, five and six phases. It is worth mentioning that, since no assumption is made for the sequence of the interface layers in the formulation of both micromechanics models, the predicted overall properties are only function of the properties of each phase and geometry of the inclusion (aspect ratio and volume fractions). Thus alternating the sequences of the layers with constant volume fractions does not affect the predicted overall electromechanical properties of the Multi-Inclusion.

6. Carbon fiber model in the presence of ZnO nanowires interphase region With the finite element analysis demonstrating the accuracy of the Multi-Inclusion model, the model can be applied to a case study to predict the effective electroelastic properties of a fiber surrounded by vertically aligned nanowires which create an interface of graded properties. For this analysis the ZnO nanowires reported by Lin et al. [37] have been analyzed and used to generate the graded properties in the interface. The nanowires can be grown in a controlled manor such that a diameter ranging from 50 to 200 nm and lengths up to 4 lm can be achieved, which necessitates the use of a model to predict the optimal interfacial conditions to maximize a particular property of interest. The presence of the ZnO nanowires creates an interphase region consisting of two blended phases, namely the ZnO and polymer, as can be seen in Fig. 5. The interface is termed a graded interphase since a discrete transition does not exist between each material but rather the volume fraction gradually changes from one phase to the next [37]. It was shown by Galan et al. [4] that the morphology of this interphase plays a critical role in the interface strength and should be important for the constitutive properties as well. The simplest method to model this three-phase composite is to model the system as three distinct layers corresponding to the carbon fiber, the blended interphase and surrounded by the matrix material. This approach corresponds to the double inclusion model, which is suited for modeling three-phase composites, yet is not able to discretize the graded region into multiple phases for a more representative approximation of the interphase properties. In order to fully understand the mechanism of this behavior and investigate

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Fig. 3. Effective elastic properties with respect to ASF volume fraction; (a) longitudinal Young’s modulus, (b) transverse Young’s modulus, (c) transverse shear modulus, (d) longitudinal shear modulus.

the overall piezoelectric properties of the interface the MultiInclusion model described in the prior sections can be applied to this problem. To model the gradient of the interphase accurately, the interface region in Fig. 5 is analyzed via image analysis software to determine the volume fraction of each phase as a function of the radial distance from the surface of the fiber. The percentage of the ZnO nanowires for each distance from the carbon fiber surface is plotted versus normalized length in the radial direction in Fig. 6a. The ZnO volume fraction can be related to the reciprocal of the radius (1/r). High concentration of the ZnO nanowires in the region close to the surface of the carbon fibers and low concentration far from the carbon fiber is consistent with the distribution of carbon nanotubes volume fraction around the carbon fiber identified by Seidel et al. [38,39] determined the effective elastic properties for carbon nanotube reinforced composites through the Mori– Tanaka and finite element models. Here, the ZnO nanowires in the interfacial region are assumed to be one phase of the polymer composite. Based on this distribution of the ZnO nanowires the interphase can be discretized into a desired number of layers with Fig. 6b showing the portion of the ZnO and polymer for the case of 15 layers. The lower portion (red) of each bar in this figure represents the volume fraction of the ZnO and the upper portion (green) shows the epoxy volume fraction. In this case a carbon fiber is in the center surrounded by 15 homogenized layers of ZnO-epoxy

and an infinite volume for polymer matrix with zero volume fraction of ZnO, thus producing 17 phases. To investigate the constitutive properties of a carbon fiber coated with vertically aligned ZnO nanowires with the MultiInclusion technique, the effective properties of each layer consisting of a specific volume fraction of ZnO and polymer must be determined. Thus, the Mori–Tanaka model is used to determine the effective ZnO-polymer properties. It should be noted that the volume fraction of the ZnO is calculated by averaging along the radius of the plotted interphase profile in Fig. 6a. The typical values for the electroelastic properties of the ZnO and polymer are provided in Table 1. For each layer in this simulation the volume fraction of ZnO in the polymer is first determined then the constitutive properties for that particular volume fraction are calculated via the Mori–Tanaka method detailed before. Finally, the homogenized properties of the ZnO–Polymer phases are used in the Multi-Inlcuison model and the overall properties for three different aspect ratios (30%, 50%, and 65%) are studied. The diameter and volume fraction of the carbon fiber is assumed to be approximately 4.5 lm and 15% respectively thus the aspect ratio shouldnot exceed 65% for cylindrical multi-layer composites to have a considerable amount of polymer as infinite matrix around the inclusion. Galan et al. [4] performed interfacial tests to demonstrate that the nanowire length is critically important to the strength of the composite. To investigate how the length of the nanowires affects

M.H. Malakooti, H.A. Sodano / Composites: Part B 47 (2013) 181–189

187

Fig. 4. Effective electrical properties with respect to ASF volume fraction: (a) relative transverse permittivity, (b) relative longitudinal permittivity, (c) longitudinal piezoelectric coupling, (d) transverse piezoelectric coupling.

Fig. 5. SEM micrograph of embedded ZnO nanowires grown vertically on the carbon fiber with an epoxy matrix wetting the nanowires at two scales; (a) 4 lm (b) 1 lm.

the simulation, three different aspect ratios are modeled. As detailed before, the aspect ratio is defined as the ratio of the thickness of the interface between the carbon fiber’s surface and the pure polymer phase to the distance from the center of the fiber to the pure polymer phase. In this case study, the ZnO nanowires grown vertically on the surface of the carbon fiber and the tip of the

nanowires is where the pure phase of polymers start since there is no ZnO phase after this point, thus the length of the ZnO nanowires over the sum of nanowires length and the radius of the fiber perfectly presents the aspect ratio. In order to determine how many layers are sufficient for modeling the interface, seven Multi-Inlcusion models for each of the three aspect ratios are

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(a) 100

(b) 100

Epoxy ZnO

90

90

ZnO percentage in the interphase

80

Interphase composition

80 70 60 50

70 60 50 40 30 20

40 10

30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

2

4

6

8

10

12

14

Number of layers along fiber radius

Normalized thickness of the interphase

Fig. 6. (a) Averaged profile of ZnO percentage along fiber radius. (b) Interphase composition for 15 layers forming the inclusion.

Aspect ratio = 0.50 Aspect ratio = 0.65

8.45 8.35 6.10 6.05 6.00 5.95 4.865 4.830 4.795 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Number of layers Fig. 8. Relative permittivity of the RVE versus number of layers.

6.85

Aspect ratio = 0.30

(b) 6.80

Aspect ratio = 0.30

Aspect ratio = 0.50

6.75

Aspect ratio = 0.50

126 121

Aspect ratio = 0.65

120

Young's modulus (GPa)

Aspect ratio = 0.30

8.65 8.55

117 114 76 74 72 52

d 3 3 Piezoelectric coupling (pC/N)

(a)

8.75

κ 33 Relative permittivity

simulated with the interphase descritized into one, two, three, five, six, 10 and 15 layers. For the case of a single layer, the interface is modeled as a new phase with effective properties of the homogenized ZnO and polymer region. The longitudinal Young’s modulus, piezoelectric coupling, and relative permittivity are plotted in Figs. 7 and 8. As these plotted properties reveal, the predicted values are strongly sensitive to the value of the aspect ratio. Hence, increasing the length of the nanowires which is represented with aspect ratio, each of the electroelastic properties of the composite increases. The gap between each aspect ratio demonstrates the important role the length of ZnO nanowires play on the constitutive properties of the RVE due to the high stiffness of the ZnO over that of the polymer and the creation of a nanocomposite in each layer. It should be noted that the vertical axis in Figs. 7 and 8 has been broken to facilitate the visualization of each curve on a single plot. This analysis highlights the importance of the Multi-Inclusion approach since the results do not converge to a consistent value until the interphase is descritized into at least six separate layers. The error of the simulated RVE with three phases is between 3% and 9% for different aspect ratios and properties. For the lower aspect ratio the predicted

Aspect ratio = 0.65

6.70 6.65 6.60 4.70 4.65 4.60 4.55 2.40 2.35

51

2.30 1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Number of layers

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Number of layers

Fig. 7. The overall electroelastic properties versus number of layers. (a) Young’s modulus. (b) Piezoelectric coupling. The vertical axis is broken to show the trend for each aspect ratio.

M.H. Malakooti, H.A. Sodano / Composites: Part B 47 (2013) 181–189

values are less affected, however the graded interface effect is critically important when the graded interface occupies a considerable volume of the RVE. 7. Conclusion Modeling of multifunctional materials is complicated due to their coupled behavior between dissimilar components. Recent developments in multifunctional materials have used functionally graded interfaces in fiber reinforced composites to achieve both strength gains as well as the incorporation of piezoelectric properties. While studies have revealed that the graded phase in the functional materials leads to improvement in the material’s performance, a comprehensive model would be advantageous for the materialists to predict the overall properties. In this study, the complexity of the graded interface between two continuous media is solved by discretizing the interface region into a large number of homogenized phases. The modeling of multiphase composites with piezoelectric properties is performed by extension of the Mori–Tanaka and Multi-Inclusion micromechanics models for the coupled behavior. By means of these two models, the overall electroelastic properties of a four-phase active structural fiber are predicted. Validation of the predicted results for the overall electroelastic properties through the finite element analysis demonstrated the accuracy of the extended models. The final results show that the extended Multi-Inclusion simulation for the effective properties converges after approximately 6 inclusions independent of the thickness of the nanowire interphase. Without the discretization of the interphase region the error is between 3% and 9% with large error as the interphase thickness increases. The effect of the functionally graded interface is important to the constitutive properties since the nanocomposite formed in the blended region possesses higher stiffness than the matrix itself. The model developed here provides a method with low computational cost to theoretical model the constitute properties of fibers containing a functionally graded interphase. Acknowledgements The authors gratefully acknowledge support from the National Science Foundation (Grant # CMMI-0846539) and the Air force Office of Scientific Research directed by Dr. Joycelyn Harrison under award #FA9550-09-1-0356. References [1] Sodano HA, Inman DJ, Park G. A review of power harvesting from vibration using piezoelectric materials. Shock Vib Dig 2004;36(3):197–205. [2] Sodano HA, Park G, Inman DJ. Estimation of electric charge output for piezoelectric energy harvesting. Strain 2004;40(2):49–58. [3] Sodano HA, Inman DJ, Park G. Comparison of piezoelectric energy harvesting devices for recharging batteries. J Intelli Mater Syst Struct 2005;16(10):799–807. [4] Galan U, Lin Y, Ehlert GJ, Sodano HA. Effect of ZnO nanowire morphology on the interfacial strength of nanowire coated carbon fibers. Compos Sci Technol 2011;71(7):946–54. [5] Dunn ML, Taya M. Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. Int J Solids Struct 1993;30(2):161–75. [6] Taya M. Electron compos 2005. [7] Chen T. Micromechanical estimates of the overall thermoelectroelastic moduli of multiphase fibrous composites. Int J Solids Struct 1994;31(22):3099–111. [8] Li JY. Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. Int J Eng Sci 2000;38(18):1993–2011. [9] Lee J, Boyd IV JG, Lagoudas DC. Effective properties of three-phase electromagneto-elastic composites. Int J Eng Sci 2005;43(10):790–825.

189

[10] Hori M, Nemat-Nasser S. Double-inclusion model and overall moduli of multiphase composites. Mech Mater 1993;14(3):189–206. [11] Hori M, Nemat-Nasser S. Double-inclusion model and overall moduli of multiphase composites. J Eng Mater Technol 1994;116(3):305–9. [12] Dunn ML, Ledbetter H. Elastic moduli of composites reinforced by multiphase particles. J Appl Mech 1995;62(4):1023–8. [13] Feng X, Fang D, Soh A, Hwang K. Predicting effective magnetostriction and moduli of magnetostrictive composites by using the double-inclusion method. Mech Mater 2003;35(7):623–31. [14] Li JY. On micromechanics approximation for the effective thermoelastic moduli of multi-phase composite materials. Mech Mater 1999;31(2):149–59. [15] Li JY. Thermoelastic behavior of composites with functionally graded interphase: a multi-inclusion model. Int J Solids Struct 2000;37(39):5579–97. [16] Eshelby JD. The Determination of the elastic field of an ellipsoidal inclusion, and related problems. Pro R Soc Lond Ser A Math 1957;241(1226):376. [17] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials. Second revised edition. Amsterdam, New York: Elsevier, 1999. [18] Schjødt-Thomsen J, Pyrz R. The Mori–Tanaka stiffness tensor: diagonal symmetry, complex fiber orientations and non-dilute volume fractions. Mech Mater 2001;33(10):531–44. [19] Tan P, Tong L. Micro-electromechanics models for piezoelectric-fiberreinforced composite materials. Compos Sci Technol 2001;61(5):759–69. [20] Odegard GM. Constitutive modeling of piezoelectric polymer composites. Acta Mater 2004;52(18):5315–30. [21] Lin Y, Sodano HA. A double inclusion model for multiphase piezoelectric composites. Smart Mater Struct 2010;19(3):035003. [22] Lin Y, Sodano HA. Concept and model of a piezoelectric structural fiber for multifunctional composites. Compos Sci Technol 2008;68(7–8):1911–8. [23] Barnett DM, Lothe J. Dislocations and line charges in anisotropic piezoelectric insulators. J Phys Status Solidi (b) 1975;67(1):105–11. [24] Dunn ML, Wienecke HA. Inclusions and inhomogeneities in transversely isotropic piezoelectric solids. Int J Solids Struct 1997;34(27):3571–82. [25] Kuo W, Huang JH. On the effective electroelastic properties of piezoelectric composites containing spatially oriented inclusions. Int J Solids Struct 1997;34(19):2445–61. [26] Gaudenzi P. On the electromechanical response of active composite materials with piezoelectric inclusions. Comput Struct 1997;65(2):157–68. [27] Berger H, Kari S, Gabbert U, Rodriguez-Ramos R, Guinovart R, Otero JA, et al. An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int J Solids Struct 2005;42(21– 22):5692–714. [28] Sun CT, Vaidya RS. Prediction of composite properties from a representative volume element. Compos Sci Technol 1996;56(2):171–9. [29] Xia Z, Zhang Y, Ellyin F. A unified periodical boundary conditions for representative volume elements of composites and applications. Int J Solids Struct 2003;40(8):1907–21. [30] Berger H, Kari S, Gabbert U, Rodríguez-Ramos R, Bravo-Castillero J, GuinovartDíaz R. A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric fibre composites. Mater Sci Eng : A 2005;412(1–2):53–60. [31] Chen XL, Liu YJ. Square representative volume elements for evaluating the effective material properties of carbon nanotube-based composites. Comput Mater Sci 2004;29(1):1–11. [32] Kari S, Berger H, Gabbert U, Guinovart-Dıaz R, Bravo-Castillero J, RodrıguezRamos R. Evaluation of influence of interphase material parameters on effective material properties of three phase composites. Compos Sci Technol 2008;68(3–4):684–91. [33] Chan HLW, Unsworth J. Simple model for piezoelectric ceramic/polymer 1–3 composites used. IEEE Trans Ultrason Ferroelectr Freq Contr 1989;36(4):434–41. [34] Lin Y, Sodano HA. Electromechanical characterization of a active structural fiber lamina for multifunctional composites. Compos Sci Technol 2009;69(11– 12):1825–30. [35] Anastasia HM. A micromechanical formulation for piezoelectric fiber composites with nonlinear and viscoelastic constituents. Acta Mater 2010;58(9):3332–44. [36] Hu GK, Weng GJ. The connections between the double-inclusion model and the Ponte Castaneda–Willis, Mori–Tanaka, and Kuster–Toksoz models. Mech Mater 2000;32(8):495–503. [37] Lin Y, Ehlert G, Sodano HA. Increased interface strength in carbon fiber composites through a ZnO nanowire interphase. Adv Funct Mater 2009;19(16):2654–60. [38] Seidel GD, Lagoudas DC, Frankland SJV, Gates TS. Micromechanics modeling of functionally graded interphase regions in carbon nanotube–polymer composites. AIAA 2006. [39] Seidel GD, Lagoudas DC. Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites. Mech Mater 2006;38(8– 10):884–907.