Time-dependent response of active composites with thermal, electrical, and mechanical coupling effect

International Journal of Engineering Science 48 (2010) 1481–1497

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Time-dependent response of active composites with thermal, electrical, and mechanical coupling effect Anastasia Muliana ⇑, Kuo-An Li Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, United States

a r t i c l e

i n f o

Article history: Available online 26 September 2010 Dedicated to Prof. K.R. Rajagopal on his 60th birthday. Keywords: Active fiber composites Piezoelectric Viscoelastic Micromechanical model Thermo-electro-mechanical coupling Conduction of heat

a b s t r a c t The mechanical and physical properties of materials change with time. This change can be due to the dissipative characteristic of materials like in viscoelastic bodies and/or due to hostile environmental conditions and electromagnetic fields. We study time-dependent response of active fiber reinforced polymer composites, where the polymer constituent undergoes different viscoelastic deformations at different temperatures, and the electromechanical and piezoelectric properties of the active fiber vary with temperatures. A micromechanical model is formulated for predicting effective time-dependent response in active fiber composites with thermal, electrical, and mechanical coupling effects. In this micromechanical model limited information on the local field variables in the fiber and matrix constituents can be incorporated in predicting overall performance of active composites. We compare the time-dependent response of active composites determined from the micromechanical model with those obtained by analyzing the composites with microstructural details. Finite element (FE) is used to analyze the composite with microstructural details which allows quantifying variations of field variables in the constituents of the active composites. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Multifunctional structures are appealing for the development of future in-service and cost-effective adaptive structures, energy conversion devices, structural health-monitoring systems, high-speed vehicles, actuators, and sensors. For example, a shape-changing capability in turbine blades would increase efficiency and prolong lifetime of the turbines by mitigating some of the effects of fatigue due to vibration and thermal stress/strain from temperature changes. To achieve multi-tasking performance and to adapt to external stimuli, multifunctional structures will require advanced materials that can survive in hostile environments, sustain mechanical loading, and incorporate transducers and electronics devices. One example of forming multifunctional structures is by bonding piezoelectric ceramics, such as lead zirconate titanate (PZT), to load bearing structural components. Piezoceramics generally have higher electro-mechanical and piezoelectric properties than piezoelectric crystals and polymers, making them capable of sustaining a wider range of loadings. A brittle characteristic of piezoceramics, however, may lead to early failure that limits their applications. PZT fibers dispersed in ductile polymers provide more flexible and compliant transducers compared to the monolithic PZT wavers [4], while maintaining high electro-mechanical and piezoelectric properties. The existence of polymer matrix in active fiber composites prevents catastrophic failure due to fiber breaking [4,23]. PZT fiber/epoxy composites are polarized

⇑ Corresponding author. Tel.: +1 979 458 3579. E-mail address: [email protected] (A. Muliana). 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.08.014

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along the axial fiber direction as opposed to the through thickness polarization in piezoceramic wafers. Thus, PZT fiber composites utilize the strongest piezoelectric constant (d333), while wafers generally use the smaller piezoelectric component (d133). PZT ceramics can exhibit time-dependent response under a coupled electro-mechanical effect [8,5,34,11,12], and polymer matrix is known for its viscoelastic characteristic especially at elevated temperatures. Different thermo-electromechanical behaviors in the fiber and matrix constituents can lead to complex response in active polymer matrix composite (PMCs). Understanding time-dependent and field coupling effects becomes important to improve reliability in using active fiber polymer composites. There have been several studies on understanding electro-mechanical coupling performance of active PMCs having unidirectional piezoelectric fibers, e.g., Williams [37–39], Wickramasinghe and Hagood [36], Williams et al. [37,40]. Active PMCs show a nonlinear stress–strain relation at relatively high strains and their overall response is frequency dependent. Williams [38] showed that the elastic constants and coefficient of thermal expansion (CTE) of active fiber composites depend on temperatures. Mahato and Maiti [20] analyzed response of smart composites, having active fiber composite transducers, at various temperatures and moisture conditions. In the above studies, active PMCs are treated as homogeneous bodies with respect to their thermo-electro-mechanical response and the entire composite bodies undergo uniform temperature and moisture changes. The performance of active composites at different, but fixed environmental conditions was examined. It is also interesting and essential to understand response of active PMCs undergoing nonuniform temperature and moisture changes throughout the composite bodies, and these changes also vary with time. Studies on thermo-electro-mechanical response due to heat conduction (or moisture diffusion) in active PMCs are currently lacking. In order to adequately predict the overall performance of composites, it is often necessary to take into account the heterogeneity in the composites. Micromechanical models have been developed to determine effective properties of active composites. Pedersen [31] discussed a generalization of a volume averaging scheme to determine effective properties of composites with coupled mechanical and nonmechanical effects, such as thermo-mechanics, piezoelectric, and electromagnetic. Dunn [6], Dunn and Taya [7], Aboudi [3], Odegard [30], Lee et al. [16], Muliana [26], among others have used micromechanical models for predicting linear properties of active composites. Odegard [30] examined electro-mechanical properties of four active PMCs, i.e., graphite/PVDF fiber composite, SiC/PVDF particulate composite, PZT-7A/LaRC-Si fiber composite, and PZT-7A/LaRC-Si particulate composite. The use of LaRC-Si (polyimide) polymer matrix is appealing for high temperature applications. LaRC-Si, however, shows pronounced viscoelastic behaviors at elevated temperatures [29]. Understanding time-dependent effect on the electro-mechanical behaviors of active composites is currently limited. Li and Dunn [17] and Jiang and Batra [14] presented a micromechanical models for predicting electro-mechanical properties of linear viscoelastic piezocomposites. They combined the correspondence principle and Mori-Tanaka micromechanical model to determine complex properties of composites. Muliana [27] presented a micromechanical model to predict nonlinear response in piezocomposites during cyclic electric field. The nonlinear response is due to polarization reversal behavior of PZT fibers and viscoelastic matrix. Li and Muliana [18] studied the effects of fiber geometries and viscoelastic matrix on the elastic, dielectric, and piezoelectric properties of active fiber composites (AFC) and macro fiber composites (MFC). The AFC has a circular fiber cross-section and the MFC has a rectangular fiber cross-section. They concluded that the effect of fiber cross-section on the electro-mechanical properties of the piezocomposites along the longitudinal fiber direction is negligible, while its effect on the properties in the transverse fiber directions is more significant. The viscoelastic matrix becomes more prominent for predicting long-term response of active composites. This study examines the time-dependent and field coupling effects on the overall performance of active PMCs consisting of PZT fibers and polymer matrix. A simplified micromechanical model having four fiber and matrix subcells is formulated to obtain effective response of the active PMCs. This approach homogenizes the composites and limited information on the local field variables in the fiber and matrix constituents can be incorporated in predicting overall response of active PMCs. We also analyze time-dependent response of active PMCs by considering more realistic microstructures of composites. Representative volume elements (RVEs) of composite microstructures having several fibers are generated using finite element (FE),1 which allows quantifying detailed variations of field variables in the constituents of the active composites. The time-dependent response obtained from the simplified micromechanical model is compared to those of the RVEs. Two types of time-dependent behaviors are considered. The first type is due to the viscoelastic matrix at various isothermal conditions. The second type is due to transient heat conduction in an active composite body where the composite body is exposed to different nonuniform temperature profiles at various instants of time until steady state is reached. This manuscript is organized as follows. Section 2 discusses general constitutive material models for the constituents in active PMCs. Section 3 presents a simplified micromechanical model for active PMCs having a unidirectional fiber reinforcement. RVE models for active PMC with roughly 20% and 40% fiber volume contents are also generated using FE. Numerical results on the overall time-dependent response of active PMCs are discussed in Section 4. Section 5 is dedicated to concluding remarks.

1 We considered RVEs having multiple fibers with circular cross-section uniformly distributed in a polymer matrix. The regular fiber arrangements are commonly found in unidirectional short fiber piezocomposites often referred as 1–3 fiber composites. The fiber cross-section is usually not perfectly circular and its dimension often varies for different fibers. To reduce the complexity, we assume all fibers in the RVEs have perfectly circular cross-section and uniformly distributed in polymer matrix and we consider RVEs as ‘numerically realistic microstructures.’

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2. Linearized constitutive material models with field coupling and time effects This section briefly discusses constitutive models for piezoelectric2 fibers and polymer matrix. The constitutive models are suitable for continuous and homogeneous bodies and are based on thermodynamics and classical mechanics approaches, neglecting the effects of crystal structures and/or molecular structures on the overall response of the bodies. The thermal, mechanical, and electrical behavior of a continuum is described in terms of the following field variables: stress r, strain e, electric field E, electric flux (displacement) D, temperature T, and entropy S. This gives a total of 20 components of field variables. These field variables are not all independent and they are related by material constants. Different sets of independent field variables can be chosen for each of the thermal, mechanical, and electrical components, resulting in a total of 10 components of independent field variables. A linear constitutive material model with a thermo-electro-mechanical coupling effect3 is written as:

eij ¼ Sijkl rkl þ dkij Ek þ aij T Di ¼ dijk rkl þ jik Ek þ pi T S ¼ akl rkl þ pk Ek þ gT

ð1Þ

where S is the elastic compliance measured at constant electric field and temperature,4 j is the permittivity at constant stress and temperature, and g is the thermal property5 at constant stress and electric field. The piezoelectric constants d measured at constant electric field and temperature are identical to the ones at constant stress and temperature and likewise for the coefficient of thermal expansion (CTEs) a and pyroelectric constants p. The initial condition for stress and electric field is usually taken as zero, but the initial condition for temperature is often not zero. In such case, the third term of the constitutive model in Eq. (1) should be expressed in term of the temperature change (T  T0), where T0 is the temperature at initial condition.6 An electric current flowing through a conductor also dissipates energy which is converted into heat. Massalas et al. [21] have incorporated the effect of heat generation due to the dissipation of energy in predicting electro-mechanical response of conductive materials. In this study, to reduce the complexity in finding solutions to boundary value problems in active PMCs, we ignore the heat generation due to dissipation effect from the flow of electric current. Alternative expression for a linear coupled thermo-electro-mechanical constitutive model can be formed by taking different independent variables. When strain, electric field, and temperature are chosen as the independent field variables, the constitutive model becomes:

rij ¼ C ijkl ekl  ekij Ek  bij T Di ¼ eijk ekl þ jik Ek þ pi T S ¼ bkl ekl þ pk Ek þ gT

ð2Þ

where C is the elastic stiffness measured at constant electric field and temperature, e is the piezoelectric constant, b is the thermo-elastic constant, and j is the permittivity at constant strain and temperature (je;T ). It can be shown easily that the material constants in Eq. (1) and those in Eq. (2) are related by:

C ¼ S1 e ¼ dC r

ð3Þ e

T

j ¼ j þ ed

The above constitutive models are used for the ferroelectric fibers in active PMCs. This study deals with polymer matrix composites, and polymers generally show a pronounced viscoelastic behavior at elevated temperatures. Following a linear viscoelastic material model that is described with an analog to linear elastic response using the correspondence principle, we generalize Eq. (1) or (2) to include the effect of loading history. Discussion and restriction on using the correspondence principle for linear viscoelastic problems can be found in Wineman and Rajagopal [42] and Rajagopal and Wineman [32]. We assume that the polymer matrix is isotropic, the electro-mechanical coupling effect is absent, and the dissipation of energy during the viscoelastic deformation is negligible. The following constitutive relation for a linearized viscoelastic deformation of solids (nonaging materials) is used for the polymer matrix in piezocomposites:

etij ¼ eij ðtÞ ¼

Z 0

t

Jðwt  ws Þ

@Ssij ds þ dij @s

Z 0

t

Bðwt  ws Þ

@ rskk ds þ aij ðT t  T 0 Þ @s

ð4Þ

Di ¼ jij Ej 2

Detailed discussion on the coupled thermo-electro-mechanical constitutive models can be found in Tiersten [35], Maugin [22] and Lines and Glass [19]. The electric field can be treated as a variable that is determined by solving Maxwell’s equation or it can be treated as a constant parameter, as previously studied in Wineman and Rajagopal [41]. 4 In order to distinguish properties measured at different conditions, a superscript is usually used, i.e., SE,T. For simplicity, we shall drop the superscript of the field variables in the material constants and we will discuss the characterization of these material constants. 5 This property is related to a specific heat of a ferroelectric. 6 It is often convenient to take the initial condition at room temperature (25 °C) as linear material properties are often measured at room temperature (reference temperature). It is also possible to include the pre-stress effect in Eq. (1). 3

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Active PMC x2 x1 x3

1

2

3

4

Unit-cell model

Fig. 1. A unit-cell model for active PMCs.

Stij

where and rtkk are the scalar components of the deviatoric and volumetric stress tensors, respectively. Superscript t indicates the time variable and dij is Kronecker delta. The corresponding linear elastic Poisson’s ratio for the viscoelastic solid is assumed constant. The shear J(t) and bulk B(t) compliances follow the same creep function as the extensional compliance:

DðtÞ ¼ Dð0Þ þ

N X

   Dn 1  exp kn wt

ð5Þ

n¼1

where N is the number of Prony terms in the transient compliance, Dn is the nth coefficient of the Prony series, and kn is the nth reciprocal of retardation time that corresponds to Dn. The reduced time function in Eq. (5) can be found in Wineman and R t df 7 Rajagopal [42], and is given as wt  wðtÞ ¼ 0 aðT f ; where a(T) is the time-temperature shift factor. Knauss and Emri [15] have Þ suggested time-dependent CTE for polymer; however, for simplicity we take CTE values to be constant. 3. A simplified micromechanical model of active PMCs A simplified micromechanical model is used to determine effective time-dependent response, with thermo-electromechanical coupling effect, of active fiber reinforced PMCs. The studied active PMC consists of unidirectional ferroelectric fibers dispersed in viscoelastic polymer matrix. The microstructures of active PMCs are idealized by a periodically distributed arrays of fibers embedded in a homogeneous polymer matrix. Interphases between the fibers and matrix are assumed to be perfect. A unit-cell model consisting of four fiber and matrix subcells is generated as illustrated in Fig. 1. The geometrical representation of the proposed micromechanical model for unidirectional fiber composites is similar to Method of Cells (MOC) (see [1,2]). However, the micromechanical formulation does not follow MOC, which is defined in terms of higher order field variables in each subcell. The present micromechanical formulation is expressed in terms of average field variables in the subcells which give crude approximate solutions. Thus, for a field-dependent and time-dependent (nonlinear) constitutive material response in the subcells, a correction scheme is added to minimize errors from the linear micromechanical relations and from using average field variables instead of higher order field variables in each subcell. Haj-Ali and Muliana [9,10] and Muliana and Sawant [25] have previously used this unit-cell model for analyzing nonlinear viscoelastic behaviors of fiber reinforced PMCs. The first subcell is the fiber constituent and subcells 2, 3, and 4 represent the matrix constituent. Muliana [26,27] has extended the above unit-cell model to predict effective piezoelectric properties of active fiber composites. For conveniences in presenting basic formulations of the homogenized properties of active PMCs, the following linearized relation is used:

Z ¼ XC

ð6Þ

where C denotes the vector of the independent variables and X is the matrix of material properties. Using the constitutive relation in Eq. (1), the components of C, Z and X are:

CT ¼ fr11 ; r22 ; r33 ; r12 ; r13 ; r23 ; E1 ; E2 ; E3 ; DTg ZT ¼ fe11 ; e22 ; e33 ; e12 ; e13 ; e23 ; D1 ; D2 ; D3 ; Sg 2 3 T S d aT 6 7 X ¼ 4 d jr pT 5

a

p

ð7Þ

g

where DT = T  T0. 7 The time shift factor is a material property that influences the internal clock of the viscoelastic material and depends on the changes of environmental conditions of the materials. It is used to determine response of a material at present time and conditions by shifting the response of this material that at a reference condition supposes to occur at different time. A more detailed discussion on internal clock and time shift factor can be found in Wineman and Rajagopal [41] and Rajagopal and Wineman [33].

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The effective response of composites is formulated in terms of a volume average of field variables of the constituents in a representative unit-cell model. Each unit-cell model is divided into a number of subcells and the spatial variation of the field variables in each subcell is assumed to be uniform. The effective field variables, indicated by an over-bar, are defined as: N Z N 1X 1X ðbÞ ðbÞ CðbÞ ðxk ÞdV  V ðbÞ Ci ; i V b¼1 V ðbÞ V b¼1 N Z N 1X 1X ðbÞ ðbÞ ðbÞ Zi ¼ Z i ðxk ÞdV  V ðbÞ Z i V b¼1 V ðbÞ V b¼1

Ci ¼

k ¼ 1; 2; 3 and i ¼ 1; 2; . . . ; 10

ð8Þ ð9Þ ðbÞ

ðbÞ

Superscript (b) denotes the subcell’s number and N is the number of subcells. Variables Ci and Z i are the components of P the average field quantities in each subcell. The unit-cell volume V is defined by V ¼ Nb¼1 V ðbÞ . This study deals with viscoelastic response of the matrix constituent which requires incorporating history of loading to determine overall response of the composites. The solutions for the field variables are performed incrementally. The incremental forms of the effective and local field variables at current time are:

Cti ¼ CitDt þ DCti Z ti ¼ Z itDt þ DZ ti ðbÞ;tDt ðbÞ;t CðbÞ;t ¼ Ci þ DC i i ðbÞ;t

Zi

ðbÞ;tDt

¼ Zi

ð10Þ

ðbÞ;t

þ DZ i

where the superscript t  Dt denotes the previous time and all the field variables associated with t  Dt are considered as history variables. At each time increment Dt, we need to calculate incremental field variables at current time. A linearized micromechanical relation is imposed to the incremental parts of the field variables and a residual arouse from the linearization is corrected incrementally to satisfy both micromechanical constraints and constitutive equations for the fiber and viscoelastic matrix. For simplicity, we shall drop the superscript related to time and the rest of the incremental formulation are presented at current time t. The local average independent field variables can be expressed in terms of effective (homogenized) field variables which in the incremental form are written as: ðbÞ DCðbÞ ¼ F ij DCj i

ð11Þ

ðbÞ F ij

where is the concentration tensor for the subcell b. The linearized micromechanical relations are given in the Appendix. These micromechanical relations give exact effective response8 ofthe homogenized composites only when linear piezoelectric constitutive equations are used for all constituents (subcells) in the unit-cell model. As we use a time-dependent constitutive model for the matrix constituent that require incorporating a history of loadings in predicting effective performance of the active composites, imposing the linearized micromechanical relations for the incremental field variables could lead to residual. At every instant of time a corrector scheme is added to minimize the residual. This study uses the Newton–Raphson iterative method as a corrector scheme. ðbÞ Once the residual has been minimized, the components of the concentration matrices F ij are determined and the field variðbÞ ables DCi in each subcell due to the prescribed boundary conditions at the macro (homogenized composite) level are calðbÞ culated using Eq. (11). Finally, the constitutive equations in Section 2 are used to calculate the field variables DZ i in each subcell and the effective field variables at every instant of time are evaluated using the volume average scheme in Eqs. (8) and (9). Composites are heterogeneous materials that can exhibit discontinuities in field variables at the interphases between different constituents and nonuniform variations in field variables. The variations and discontinuities in field variables at the microscale cannot be captured sufficiently if one treats the composites as homogenous (homogenized) materials. Although the present homogenization scheme recognizes the different responses in the fiber and matrix constituents, field variables obtained using the present homogenization scheme represent approximate (average) field variables with limited spatial variations. The present micromechanical model does not quantify detailed variations in local field variables of the constituents as in the heterogeneous materials (only four subcells are considered). The simplified micromechanical model is beneficial for analyzing performance of large-scale multifunctional structures, whose components made up of active PMCs, undergoing changes in the properties of the constituents with time and external stimuli. In order to evaluate reliability of the simplified micromechanical model in predicting overall response of active PMCs, we compare its predictions with those of ‘‘numerically more realistic microstructures”. For this purpose, we generate microstructures of active PMCs using FE for composites with around 20% and 40% fiber volume contents as illustrated in Fig. 2.9 The diameter of the ferroelectric fibers is 8 The exact effective response in this context is due to satisfying the volume average scheme in Eqs. (18) and (19) and linear thermo-electro-mechanical constitutive relations of all subcells. This study assumes constant field variables in each subcell, while the exact field variables in each subcell often vary with the locations. The use of the mean field approach results in approximate solutions of the effective field variables. 9 By selecting these microstructures we made an assumption that fibers are uniformly distributed in the entire matrix medium and have the same crosssection, which generally contradicts with real microstructures of composites.

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Fig. 2. FE meshes of active PMC with 19.6% and 44.2% fiber volume contents.

Table 1 Electro-mechanical and piezoelectric properties of PZT-7A [30]. Properties

PZT-7A

LaRC-Si

E11 = E22 (GPa) E33

2.67

G12 G13 = G23

94.95 81.88 0.322 0.383 35.9 25.4

e113 = e113 (C/m2) e311 = e322 e333

9.2 2.1 9.5

je11 =jo ¼ je22 =jo je33 =jo

460 235

m12 m13 = m23

0.4

–

2.8

(jo = 8.854187816  1012 F/m).

250 lm and the volume of the represented microstructures is 1  1  1 mm3. Three-dimensional continuum elements C3D8 and C3D8E of ABAQUS are used for the polymer matrix and ferroelectric fibers, respectively. With the above fiber diameter, the RVEs with four and nine fibers have fiber volume contents of 19.6% and 44.2%, respectively. A user material subroutine (UMAT) is used to incorporate the viscoelastic constitutive model for the polymer matrix. A time-integration algorithm for the viscoelastic matrix can be found in Haj-Ali and Muliana [9,10] and Muliana and Khan [24]. 4. Effective response of active PMCs We examine average (effective) time-dependent response of active PMCs, determined using the four-cell micromechanical model. The response is compared with the one obtained using FE simulation. We first characterize the effective linear properties of PZT-7A/LaRC-Si active PMCs. The electro-mechanical properties of PZT-7A fiber10 and LaRC-Si matrix at room temperature are taken from Odegard [30] and given in Table 1. LaRC-Si is modeled as an isotropic polymer. Comparisons of the effective electro-mechanical properties determined from the simplified micromechanical model and FE analyses of active PMC with detailed microstructures are given in Table 2. It is seen that the simplified micromechanical model show a reasonably good agreement with the FE with errors (shown in parenthesis) less than 5%. Further comparison and verification of linear electro-mechanical properties can be found in Muliana [26,27] and Li and Muliana [18]. Active PMCs are often exposed to high mechanical and electrical stimuli where the driving voltage can take up to 1500 V. Under this condition, significant amount of heat could be generated, increasing the temperature of the active PMCs. At elevated temperatures, polymers show significant viscoelastic behavior. We examine time-dependent behavior of PZT-7A/ LaRC-Si composites at elevated temperatures. Creep data for the LaRC-Si matrix are obtained from Nicholson et al. [29] which show significant viscoelastic responses at temperatures: 213–223 °C (Fig. 3). We use a creep function based on the 10 Fibers are transversely isotropic with respect to their thermo-electro-mechanical response and the polarization axis of the fibers is in the axial fiber direction (x3).

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A. Muliana, K.-A. Li / International Journal of Engineering Science 48 (2010) 1481–1497 Table 2 Effective electro-mechanical properties of active PMCs with 19.6% and 44.2% fiber volume contents (at room temperature). Properties

Vf = 19.6%

Axial modulus E33 (GPa) Piezoelectric e333 (C/m2) Permittivity je33 =jo

Vf = 44.2%

Micromodel

FE

Micromodel

FE

18.2 (0.6%) 2.12 (0.9%) 49.13 (0.8%)

18.09 2.14 49.53

37.61 (0.7%) 4.76 (0.02%) 100.29 (4.3%)

37.34 4.77 95.97

Fig. 3. Creep response of LaRC-Si (experiment and Prony series fitting curve).

Table 3 Elastic compliance and Prony parameters for LaRC-Si. Temperature (°C)

D(0) (1/GPa)

D1 (1/GPa)

k1 (103 1/min)

213 218 223

0.375 0.371 0.313

0.2233 0.3603 0.4915

1.602 1.230 1.088

exponential form in Eq. (5) and we calibrate the Prony coefficients by fitting these available creep data. Only one term of the Prony series is chosen. Table 3 presents the calibrated elastic compliance and Prony parameters for the LaRC-Si matrix at three different temperatures. Limited data of the PZT properties at elevated temperatures are available, which for the jr PZT-7A11 at 218 °C are reported only for d311 ¼ 130:5  1012 C=N and j33o ¼ 1463 [13]. In this study, the rest of the piezoelectric and dielectric constants are considered to vary linearly with temperatures, from a reference temperature at 25 °C, in a similar way as the two values obtained from Jaffe and Berlincourt [13]. We assume that the elastic properties of the PZT-7A at 213–223 °C are the same as the ones at room temperature. The piezoelectric and dielectric properties at various temperatures are reported in Table 4. We monitor the time-dependent response of the active PMCs, having 19.6% and 44.2% fiber volume contents, subject to several boundary conditions. The following case studies are considered. 4.1. Time-dependent behavior due to prescribed mechanical boundary conditions We first apply a constant strain in the direction of the fiber e33 = 0.1% to the active composite for 2000 min and we monitor the corresponding stress relaxation and electric flux. The active composite is under a closed electric circuit in which the net electric field is zero. Fig. 4 illustrates the effective stress relaxation in the direction of the fiber (r33) for composites with 19.6% fiber volume contents at three isothermal temperatures. The stress relaxation is more pronounced at higher temperatures since the polymer relaxes (creeps) faster at higher temperatures. The mismatches in the properties of the fiber and matrix increase with time and temperature, resulting in larger internal stresses in the constituents. The effect of thermal 11

The Curie temperature of PZT-7A is 350 °C.

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Table 4 Piezoelectric and dielectric properties of PZT-7A. Properties

T = 25 °C

T = 213 °C

T = 218 °C

T = 223 °C

d113 = d113 (1012 C/N) d311 = d322 d333

362 53.4 133

437 128.5 208

439 130.5* 210

441 132.5 212.0

9.2 2.1 9.5

19.3 4.8 22.6

e113 = e223 (C/m2) e311 = e322 e333

jr11 =jo ¼ jr22 =jo jr33 =jo *

836 403

20 5.1 23.4

1868 1436

20.7 5.2 24.2

1896 1463*

1923 1490

Properties obtained from [13].

18.5

18.5

(b) T=218 oC

(a) T=213 oC

FE

Micromodel

18

Axial stress ( σ 33 MPa)

Axial stress ( σ 33 MPa)

FE

17.5

17

16.5

Micromodel

18

17.5

17

16.5

0

500

1000

1500

2000

0

500

Time (min)

1000

1500

2000

Time (min)

18.5

(c) T=223 oC

Axial stress ( σ 33 MPa)

FE Micromodel

18

17.5

17

16.5 0

500

1000

1500

2000

Tim e (m in) Fig. 4. Stress relaxation (r33) of PZT-7A/LaRC-Si composites with 19.6% fiber volume contents due to a constant strain e33 = 0.1%.

stress due to the different CTEs of the fiber and matrix is ignored, as we are interested only in the relaxation response due to the mechanical loadings. Response obtained from the micromechanical model is comparable to the one from the FE analysis with a constant deviation due to the mismatches in the elastic properties (Table 2). In this particular case, the electric flux D3 remains almost constant with time and its values from 213 °C to 223 °C are reported as follows (FE calculations are given in parenthesis): 0.00505 (0.00502); 0.00524 (0.00520); 0.00541 (0.00537) C/m2. Stress relaxation for the composites with 44.2% fiber contents are presented in Fig. 5. As expected increasing fiber volume contents reduces the relaxation behaviors. The electric flux for the composites with 44.2% fiber contents is shown to be independent of time and its values from 213 °C to 223 °C are reported as follows (FE calculations are given in parenthesis): 0.01135 (0.01129); 0.01178 (0.01169); 0.01219 (0.01209) C/m2. The mismatches in the overall response determined using the simplified micromechanical model and FE

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38

38

(b) T=218 oC

(a) T=213 oC

FE

Micromodel

37.5

Axial stress ( σ 33 MPa)

Axial stress ( σ 33 MPa)

FE

37

36.5

Micromodel

37.5

37

36.5

36

36 0

500

1000 Time (min)

1500

38

2000

0

500

1000 Time (min)

1500

2000

(c) T=223 oC

Axial stress ( σ 33 MPa)

FE Micromodel

37.5

37

36.5

36 0

500

1000

1500

2000

Tim e (m in) Fig. 5. Stress relaxation (r33) of PZT-7A/LaRC-Si composites with 44.2% fiber volume contents due to a constant strain e33 = 0.1%.

analysis of composites with microstructural details increase as fiber volume content of the composite increases. The deviations are between 0.4% and 2.8%. As the fiber volume content increases the distant (spacing) between the fibers is smaller; thus, nonuniformity in the stresses around the fiber spacing, which is due to discontinuities in the field variables, increases. The simplified micromechanical model is formulated in terms of average field variables in the four fiber and matrix subcells, and as a result, limited stress variations in the constituents can be captured. Now, we examine the relaxation response at different temperatures when the effect of thermal stresses due to the mismatches in the CTEs of the fibers and matrix are considered. The CTEs for the fibers are a11 ¼ a22 ¼ 5  106 = C and a33 ¼ 5  106 = C, and the one of the isotropic matrix is a11 ¼ a22 ¼ a33 ¼ 50  106 = C. Fig. 6 illustrates the stress relaxation in the direction of the fiber (r33) for composites with 19.6% fiber volume contents. Incorporating the thermal stresses shifts the stress levels, but insignificantly affects the relaxation behaviors. Table 5 presents changes in the axial stresses after 2000-min relaxation. We then apply a constant stress in the axial fiber direction (r33 = 10 MPa) for 2000 min and we monitor the corresponding axial strain, transverse strain, and electric flux, as reported in Figs. 7 and 8. Like in the stress relaxation, a closed electric circuit condition is imposed during the creep deformation. Only the composite with 19.6% fiber volume contents is considered and the effect of the thermal stresses is ignored. The effect of viscoelastic matrix on the overall strain and electric flux is shown to be significant even though the composites are loaded in their axial fiber direction. The overall time-dependent response of active PMCs determined from the simplified micromechanical model and FE is in a good agreement. Nonuniform strain and stress fields in the composite having microstructural details are illustrated in Fig. 9. Responses are reported at two different times: 50 and 960 min. It is seen that more pronounced variations in the stress and strain fields are shown in the locations near fiber–matrix interphases. These variations could have significant influence on the overall performance of composites, whereas the simplified micromechanical model can only incorporate four different variations in the field variables. It is interesting to observe that during the stress relaxation the electric flux is independent of time, while the electric flux shows time-dependent behavior during the creep. In both relaxation and creep problems, a net of electric field is equal to zero. When a uniform and constant strain is prescribed in the axial fiber direction, the fiber and matrix constituents exhibit different stresses and it is not necessary to impose a uniform stress condition in the axial fiber direction. As a result, the

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20

20

(b) T=218oC

(a) T=213oC

19.5 Axial stress (σ 33 MPa)

Axial stress (σ 33 MPa)

19.5 19 18.5 With CTE 18

Without CTE

17.5

19 18.5 With CTE 18

Without CTE

17.5 17

17

16.5

16.5

0

500

1000

1500

2000

0

500

Time (min)

1000

1500

2000

Time (min) 20 (c) T=223 oC

Axial stress (σ 33 MPa)

19.5 19 18.5 With CTE 18

Without CTE

17.5 17 16.5

0

500

1000

1500

2000

Time (min) Fig. 6. Stress relaxation (r33) of PZT-7A/LaRC-Si composites with 19.6% fiber volume contents due to a constant strain e33 = 0.1% with and without thermal stresses.

Table 5 Amount of stress relaxation after 2000 min for composites with 19.6% fiber volume contents. Temperature (°C)

213 218 223

Dr33 = r33(2000)  r33(0) MPa With CTE

Without CTE

0.791 1.050 1.545

0.90 1.048 1.554

effective electric flux is related directly to the applied constant strain. When a uniform and constant stress is prescribed in the axial fiber direction, the fiber and matrix constituents exhibit different normal strains and to maintain a perfect bond between the fiber and matrix it is necessary to impose a uniform displacement condition in the axial fiber direction. Imposing a uniform displacement condition, while at the same time the mechanical properties of the matrix changes over time, requires adjusting stresses in the fiber and matrix constituents continuously. Although the overall stress is fixed to a constant value, stresses exhibited in the fiber and matrix would vary with time, resulting in time-dependent electric flux. Since the stresses in the matrix relax with time, fibers would have to experience increase in their internal stresses with time to maintain an equilibrium condition (balance of linear momentum). The electro-mechanical coupling effect in the fiber leads to increase in the overall electric flux with time. Active PMCs are generally used as sensors or actuators. As a sensor, the corresponding electric charge (or electric field) due to deformation in a structure is continuously monitored with time, while in an actuator application, one need to supply a certain amount of electric charge (or electric potential) to control deformation. When the electro-mechanical coupling behaviors of active PMCs are time-dependent, it is then essential to take into account a history of loadings in the active PMCs in interpreting the electric feedback from a sensor or in supplying electric charge (or electric potential) to control the deformation.

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0.06

0

0.056 0.054

FE Micromodel

0.052

Transverse strain ( ε11 %)

0.058 Axial strain ( ε33 %)

Tim e (m in)

-0.02

(a) T=213 oC

500

1000

1500

2000

-0.021

-0.022

-0.023 FE

-0.024

(d) T=213 oC

Micromodel

0.05 0

500

1000

1500

2000

-0.025

Time (min)

0.06 0

0.056

0.054

FE Micromodel

0.052

Transverse strain ( ε11 %)

0.058 Axial strain ( ε33 %)

Tim e (m in)

-0.02

(b) T=218 oC

500

1000

1500

2000

-0.021

-0.022

-0.023 FE

-0.024

(e) T=218 oC

Micromodel

0.05 0

0.06

500

1000 Time (min)

1500

2000

-0.025

(c) T=223oC

0

0.056

0.054

FE Micromodel

0.052

Transverse strain ( ε11 %)

0.058 Axial strain ( ε33%)

Tim e (m in)

-0.02 500

1000

1500

2000

-0.021

-0.022

-0.023

-0.024

FE (f) T=223 oC

Micromodel

0.05 0

500

1000

1500

2000

-0.025

Tim e (m in) Fig. 7. Axial creep strain (e33) and transverse creep strain (e11) of PZT-7A/LaRC-Si composites with 19.6% fiber volume contents due to a constant stress r33 = 10 MPa.

4.2. Time- and field coupling behavior due to conduction of heat Response of active composites can vary with time due to changes in their environmental conditions. For example, diffusion of a fluid and/or conduction of heat alter the mechanical and physical properties of the composites. In this study, we examine response of field variables, i.e., temperature, displacement, stress, electric field, and electric flux, in active PMCs during transient heat conduction throughout the composite body. We use the simplified micromechanical model to determine overall field variables in the composites. The advantage of using micromechanical models to determine effective properties and calculate the overall response of the composites, as compared to assigning fixed values of the effective properties, is that

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0.2 Electric flux (D3 10-3 C/m 2)

Electric flux (D3 10-3 C/m 2)

0.2 0.19 0.18 (a) T=213 oC

0.17

FE Micromodel

0.16 0.15

0.19

0.18 (b) T=218 oC 0.17

FE Micromodel

0.16 0.15

0

500

1000 Time (min)

1500

2000

0

500

1000 Time (min)

1500

2000

Electric flux (D3 10-3 C/m 2)

0.2

0.19

0.18

(c) T=223oC

0.17

FE Micromodel

0.16

0.15 0

500

1000 Time (min)

1500

2000

Fig. 8. Electric flux along the polarization axis (D3) of PZT-7A/LaRC-Si composites with 19.6% fiber volume contents due to a constant stress r33 = 10 MPa.

it allows incorporating different time- and field dependent behaviors for the constituents during the conduction of heat and deformation in the composite body. We consider an active fiber composite panel where the longitudinal fibers are aligned in the x3 direction undergoing conduction of heat along the longitudinal fiber axis (Fig. 10). It is assumed that the composite panel is perfectly insulated so that the amount of heat loss is ignored. The one-dimensional conduction of heat follows Fourier’s law and is written as:

Hc

@T @  @T k33 ¼ @t @x3 @x3

! ð12Þ

33 are the effective heat capacity and component of the effective thermal conductivity in the axial fiber direcwhere Hc and k tion. In general, the thermal conductivities of the fiber and matrix constituents are temperature dependent, resulting in the effective thermal conductivity of the composites being dependent on the temperature field as well. Muliana and Kim [28] have presented a micromechanical formulation for analyzing transient heat conduction and obtaining effective thermal properties of unidirectional fiber reinforced composites. The thermal properties of the constituents, i.e., heat capacity and thermal conductivity, are allowed to vary with temperature field and indeed with time. The energy equation (Eq. (12)) can be modified to incorporate heat generation due to the dissipation effects. In this study, we use the micromechanical model presented in Muliana and Kim [28] to determine temperature field in the composite panel subject to the following initial and boundary conditions:

Tðx1 ; x3 ; 0Þ ¼ 25  C 80 6 x1 6 1;

0 6 x3 6 1

Tðx1 ; 1; tÞ ¼ 213  C 80 6 x1 6 1;

t P 0:0

@Tðx1 ; 0; tÞ ¼ 0:0 80 6 x1 6 1; @x3

ð13Þ

t P 0:0

The thermal properties for the fiber (PZT material) are k11 ¼ k22 ¼ k33 ¼ 1:1 W=m  C and Hc ¼ 2:73  106 J=m3  C, while the thermal properties for the matrix (epoxy material) are k11 ¼ k22 ¼ k33 ¼ 0:19 W=m  C and Hc ¼ 1:47  106 J=m3  C. In

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Fig. 9. Strain and stress fields in composite with microstructural details during creep response (the first column is response at t = 50 min and the second column is response at t = 960 min).

x1

1 mm

1 mm

x3 Fig. 10. Two-dimensional model of active PMCs.

order to simplify the solutions, these properties are assumed to be independent on temperatures. Fig. 11a illustrates the temperature profile for the active PMCs with 19.6% fiber volume content. The composite panel of 1  1 mm dimension is chosen to accelerate the conduction process. We analyze the electro-mechanical response of the active composite panel due to the thermal stress and thermo-electro coupling effects during the transient heat conduction. The panel is constrained along its edges, except at the edge that is in contact with the high temperature, and is under a closed electric circuit. The following boundary conditions are prescribed to the composite panel at any instant of time:

1 ð1; x3 ; tÞ ¼ 0:0 80 6 x3 6 1; 1 ð0; x3 ; tÞ ¼ u u 3 ðx1 ; 0; tÞ ¼ 0:0 80 6 x1 6 1; t P 0:0 u t 3 ðx1 ; 1; tÞ ¼ 0:0 80 6 x1 6 1; t P 0:0

t P 0:0 ð14Þ

 ðx1 ; 1; tÞ ¼ uo 80 6 x1 6 1; t P 0:0  ðx1 ; 0; tÞ ¼ u u  1 and u  3 are the components of the effective displacement, t3 is the component of the surface traction, and u  is the where u electric potential. Since the steady state in the composite panel is reached in a very short duration (less than 20 s), we neglect the effect of the viscoelastic matrix. The equations that govern the electro-mechanical response are given as:

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250

8

o

Temperature ( C)

200

150 100 t=0.08 s t=0.8 s

50

t=2.4 s

Displacement u 3 (x10-3 mm)

(a) Temperature

(b) Displacement

t=0.08 s

6

t=0.8 s t=2.4 s t=4.8 s

4

2

t=4.8 s

0 0

0.2

0.4

0.6

0.8

0 0

1

0.2

Location along x 3 (mm)

0.04

t=0.08 s

Transverse stress σ11 (GPa)

0.5

(c) Axial stress

t=0.8 s

0.4

t=2.4 s t=4.8 s

0.1 0 0

Electric flux D3 (C/m 2)

0.04

0.2

0.4 0.6 Location along x 3 (mm)

0.8

1

0.2

-0.02 t=0.08 s t=1.6 s

1

t=0.08 s t=2.4 s t=4.8 s

0.1

0.4

0.6

0.8

1

0.08

(f) Electric field t=0.08 s t=0.8 s

0.06

t=2.4 s t=4.8 s

0.04

0.02

t=4 s t=8 s

-0.06

0.4 0.6 0.8 Location along x 3 (mm)

-0.16

(e) Electric flux

Location along x 3 (mm)

-0.04

0.2

-0.12

0.02

0 0

1

t=0.8 s

-0.08

0.2

0.8

(d) Transverse stress

0 0

-0.04

0.3

Electric field E 3 (MV/m)

Axial stress σ 33 (GPa)

0.6

0.4 0.6 Location along x 3 (mm)

0 0

0.2

0.4 0.6 Location along x 3 (mm)

0.8

1

Fig. 11. Field variables of the active PMC panel during transient heat conduction process.

C 3333 e333

 3  3 @C 3333 @ u   @ e333 @ u @2u @2u 33 @T  @ b33 ðT  T o Þ ¼ 0:0 þ þ e333 2 þ b 2 @x3 @x3 @x3 @x3 @x3 @x3 @x3 @x3

 @u 3  3  3 @ e333 @ u  @j @2u @2u @T @ p  33 2  33 3 þ j þp þ ðT  T o Þ ¼ 0:0 @x3 @x3 @x3 @x3 @x3 @x3 @x23 @x3

ð15Þ

3 is the effective pyroelectric constant and the dielectric constant is measured at constant strain and temperature. where p The pyroelectric constant for the fiber is p3 ¼ 5:5  106 C=m2  C. In order to clarify the effect of thermal stress on the overall response of the composite, we neglect the temperature dependent properties, which further simplify the governing equations in Eq. (15). The analytical solutions are obtained by imposing a plane stress condition. Fig. 11(b)–(f) depicts the axial displacements, stresses, electric fluxes, and electric fields at several instants of time during the transient heat conduction. The nonuniform temperature profiles at early times lead to relatively high axial stress, electric field, and electric flux. As time

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progresses and steady state condition is reached, the axial stress and electric field should drop to zero, while the electric flux approaches a constant value. On the other hand, the transverse stress increases as time progresses and the maximum value is at the steady state condition, which is expected. 5. Conclusions We have studied time-dependent response of active PMCs comprised of unidirectional PZT fibers and epoxy matrix. Two types of time-dependent behaviors have been considered. In the first type, we have examined the effect of viscoelastic matrix at various isothermal conditions on the overall electro-mechanical response of active PMCs. We found that the time-dependent response in the composites depends not only on the properties and compositions of the constituents, but also on the prescribed boundary conditions. We have investigated creep and relaxation responses of active PMCs under a closed electric circuit. During the relaxation under a constant strain applied to the axial fiber direction the electric flux is shown to be independent of time. On the other hand, the electric flux shows more pronounced time-dependent behavior when a constant stress is prescribed in the axial fiber direction. As the stresses in the matrix relax with time under creep, fibers would have to experience increase in their internal stresses with time to maintain equilibrium condition; thus, increasing overall electric flux with time. The second type of time-dependent response is due to the effect of thermal stress and thermo-electro-mechanical coupling during transient heat conduction in an active composite. During heat conduction, the composite body is exposed to a nonuniform temperature field that also varies with time. The coupled thermo-electro-mechanical effect leads to various changes in the field variables during the conduction of heat. When the composite body is exposed to a high temperature gradient, the composite experiences significant variations in stress, electric flux, and electric field, and in some cases, these field variables show large magnitudes. As a steady state is reached, field variables approach constant values, which are expected, and at a steady state, these constant values can be smaller in magnitudes than those in the transient part. In general, material properties can vary significantly with field variables (nonlinear response); thus, severe variations in field variables throughout the composite body could potentially lead to damage initiation and failure. If we merely study the response of active PMCs at a steady state, false detection of localized damage (or failure) can occur. Acknowledgement This research is sponsored by the Air Force Office of Scientific Research (AFOSR) under Grant FA 9550-10-1-0002. Appendix A We present micromechanical relations between the fiber and matrix subcells in the unit-cell model by imposing traction continuity and displacement compatibility at the interphases of all subcells. The linearized micromechanical relations for active PMCs, having unidirectional fibers aligned in the x3 direction, are summarized as follows. The relations along the fiber axial direction (x3) are: ð1Þ ð2Þ ð3Þ ð4Þ e33 ¼ e33 ¼ e33 ¼ e33 ¼ e33 ð3Þ ð4Þ ð1Þ ð1Þ ð2Þ ð2Þ V r33 þ V r33 þ V ð3Þ r33 þ V ð4Þ r33 ¼ r33 ð1Þ

ð2Þ

ð3Þ

ð4Þ

E3 ¼ E3 ¼ E3 ¼ E3 ¼ E3 ð1Þ

ðA:1Þ

ð2Þ

ð3Þ

ð4Þ

V ð1Þ D3 þ V ð2Þ D3 þ V ð3Þ D3 þ V ð4Þ D3 ¼ D3

ðA:2Þ

The micromechanical relations in the x2 direction are: ð3Þ rð1Þ 22 ¼ r22 ð4Þ rð2Þ 22 ¼ r22

V ð1Þ V

ð1Þ

þV V ð2Þ

ð3Þ

V ð2Þ þ V ð4Þ

ð1Þ e22 þ ð2Þ e22 þ

V ð3Þ V

ð1Þ

þ V ð3Þ V ð4Þ

V ð2Þ þ V ð4Þ

eð3Þ 22 ¼ e22

ðA:3Þ

eð4Þ 22 ¼ e22

ð3Þ rð1Þ 23 ¼ r23 ð4Þ rð2Þ 23 ¼ r23

V ð1Þ V

ð1Þ

þV V ð2Þ

ð3Þ

V ð2Þ þ V ð4Þ

ð1Þ e23 þ ð2Þ e23 þ

V ð3Þ V

ð1Þ

þ V ð3Þ V ð4Þ

V ð2Þ þ V ð4Þ

eð3Þ 23 ¼ e23 eð4Þ 23 ¼ e23

ðA:4Þ

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A. Muliana, K.-A. Li / International Journal of Engineering Science 48 (2010) 1481–1497 ð1Þ

ð3Þ

D2 ¼ D2 ð2Þ

ð4Þ

D2 ¼ D2 V V

ð1Þ

ð1Þ

þV V ð2Þ

V ð3Þ

ð1Þ

ð3Þ

E2 þ

V

ð2Þ

V ð2Þ þ V ð4Þ

E2 þ

ð1Þ

þ V ð3Þ V ð4Þ

V ð2Þ þ V ð4Þ

ð3Þ

E2 ¼ E2

ðA:5Þ

ð4Þ

E2 ¼ E2

The micromechanical relations in the x1 direction are: ð2Þ rð1Þ 11 ¼ r11 ð3Þ r11 ¼ rð4Þ 11

V ð1Þ V

ð1Þ

þV V ð3Þ

ð2Þ

V ð3Þ þ V ð4Þ

ð1Þ e11 þ ð3Þ e11 þ

V ð2Þ V

ð1Þ

þ V ð2Þ V ð4Þ

V ð3Þ þ V ð4Þ

eð2Þ 11 ¼ e11

ðA:6Þ

eð4Þ 11 ¼ e11

ð2Þ rð1Þ 13 ¼ r13 ð3Þ r13 ¼ rð4Þ 13

V ð1Þ V

ð1Þ

þV V ð3Þ

ð2Þ

V ð3Þ þ V ð4Þ ð1Þ

ð1Þ e13 þ ð3Þ e13 þ

V ð2Þ V

ð1Þ

þ V ð2Þ V ð4Þ

V ð3Þ þ V ð4Þ

eð2Þ 13 ¼ e13

ðA:7Þ

eð4Þ 13 ¼ e13

ð2Þ

D1 ¼ D1 ð3Þ D1

ð4Þ

¼ D1 V ð1Þ

V

ð1Þ

þV V ð3Þ

V ð2Þ

ð1Þ

ð2Þ

V ð3Þ þ V ð4Þ

E1 þ ð3Þ

E1 þ

V

ð1Þ

þ V ð2Þ V ð4Þ

V ð3Þ þ V ð4Þ

ð2Þ

E1 ¼ E1

ðA:8Þ

ð4Þ

E1 ¼ E1

The transverse shear relations are given as: ð2Þ ð3Þ ð4Þ rð1Þ 12 ¼ r12 ¼ r12 ¼ r12 ¼ r12 ð1Þ ð2Þ ð3Þ ð4Þ V ð1Þ e12 þ V ð2Þ e12 þ V ð3Þ e12 þ V ð4Þ e12 ¼ e12

ðA:9Þ

The relations for the entropy and temperature are:

DT ð1Þ ¼ DT ð2Þ ¼ DT ð3Þ ¼ DT ð4Þ ¼ DT V ð1Þ Sð1Þ þ V ð2Þ Sð2Þ þ V ð3Þ Sð3Þ þ V ð4Þ Sð4Þ ¼ S

ðA:10Þ

References [1] J. Aboudi, Micromechanical characterization of the non-linear viscoelastic behavior of resin matrix composites, Compos. Sci. Technol. 38 (1990) 371– 386. [2] J. Aboudi, Mechanics of Composite Materials: A Unified Micromechanical Approach, Elsevier, 1991. [3] J. Aboudi, Micromechanical analysis of fully coupled electro-magneto-thermo-electro-elastic multiphase composites., Smart Mater. Struct. 10 (2001) 867–877. [4] A.A. Bent, N.W. Hagood, Piezoelectric fiber composites, J. Intell. Mater. Syst. Struct. 8 (1997) 903–919. [5] H. Cao, A.G. Evans, Nonlinear deformation of ferroelectric ceramics, J. Am. Ceram. Soc. 76 (1993) 890–896. [6] M.L. Dunn, Micromechanics of coupled electroelastic composites: effective thermal expansion and pyroelectric coefficients, J. Appl. Phys. 73 (1993) 5131–5140. [7] M.L. Dunn, M. Taya, Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites, Int. J. Solids Struct. 30 (1993) 161– 175. [8] T. Fett, G. Thun, Determination of room-temperature tensile creep of PZT, J. Mater. Sci. Lett. 17 (1998) 1929–1931. [9] R. Haj-Ali, A. Muliana, A multi-scale constitutive formulation for the nonlinear viscoelastic analysis of laminated composite materials and structures, Int. J. Solids Struct. 41 (2004) 3461–3490. [10] R.M. Haj-Ali, A.H. Muliana, Numerical finite element formulation of the schapery nonlinear viscoelastic material model, Int. J. Numer. Methods Eng. 59 (2004) 25–45. [11] D.A. Hall, Review nonlinearity in piezoelectric ceramics, J. Mater. Sci. 36 (2001) 4575–4640. [12] C. Heiling, K.H. Härdtl, Time dependence of mechanical depolarization in ferroelectric ceramics, in: Proceedings of the Eleventh IEEE International Symposium of Applications on Ferroelectrics (ISAF), 1998, pp. 503–508.

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