Average electro-mechanical properties and responses of active composites

Computational Materials Science 82 (2014) 405–414

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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Average electro-mechanical properties and responses of active composites Vahid Tajeddini a, Chien-Hong Lin a, Anastasia Muliana a,⇑, Martin Lévesque b a b

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA CREPEC, Department of Mechanical Engineering, Ecole Polytechnique de Montreal, Canada

a r t i c l e

i n f o

Article history: Received 5 June 2013 Received in revised form 19 September 2013 Accepted 23 September 2013 Available online 9 November 2013 Keywords: Particles reinforced piezoelectric composites Micromechanical models Porous piezoelectric ceramics Nonlinear electro-mechanical response

a b s t r a c t This study deals with the overall electro-mechanical response of randomly positioned spherical particles reinforced piezoelectric composites. Different composites comprising of linearly elastic and piezoelectric constituents were studied. For the piezoelectric constituent, both linear and nonlinear electro-mechanical coupling behaviors were considered. Numerical representative volume elements (RVEs) were generated and finite element (FE) method was used in order to compute overall electro-mechanical response of the RVEs. The electro-mechanical predictions of the RVEs were compared against those of Mori–Tanaka, self-consistent and simplified unit-cell micromechanical models. A new first moment secant linearization was introduced in order to perform the homogenization of the nonlinearly piezoelectric composites followed by iteration in order to minimize errors (residual) from the linearization. For all boundary conditions, including nonlinear response, simulated in this work, the predictions given by the Mori–Tanaka and UC models were reasonably close to the ones of the RVE cases. Finally the RVEs were modified to examine the linear and nonlinear electro-mechanical responses of piezoelectric ceramics with pores. Depending on the prescribed boundary conditions, the existence of pores could significantly alter the electro-mechanical response of piezoelectric ceramics. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Piezoelectric ceramics have been widely used as sensors or actuators, due to their high electrical and mechanical coupling properties. In actuator applications, piezoelectric ceramics are often subjected to large electric fields leading to nonlinear electromechanical responses. Because of their brittleness and low energy absorption capacity, piezoelectric ceramics are often mixed with ductile/soft constituents to obtain more compliant active materials. One popular type of piezoelectric composites consists of piezoelectric particles dispersed into a polymeric matrix, known as an active particulate composite. Medical ultrasound devices [1] and active damping devices [2] are two examples of industrial applications of these piezoelectric composites. Another example of active composites includes ductile polymer particles dispersed into a piezoelectric ceramic; in such example the composite microstructure can be approximated by ductile particles dispersed in a homogeneous piezoelectric matrix. Furthermore, depending on the processing methods, piezoelectric ceramics can also contain an appreciable amount of porosity that can affect their overall electro-mechanical behavior.

⇑ Corresponding author. Tel.: +1 979 458 3579; fax: +1 979 845 3081. E-mail address: [email protected] (A. Muliana). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.09.052

One approach to compute the effective properties of a heterogeneous material is using a micromechanical model, which requires identifying a representative volume element (RVE) of the system and prescribing boundary conditions. This is typically achieved by generating a large number of statistical realizations of the random microstructures, referred to as volume elements (VEs) herein, and confronting an RVE definition criterion. The heterogeneous medium can be considered statistically homogeneous when the RVE criterion is met and the effective field variables are subsequently calculated by volume averaging of the local field variables over the RVE. Several studies were devoted to defining RVEs with detailed or more realistic microstructural morphologies. For example, Gusev [3], Michel et al. [4], Segurado and Llorca [5] and Ghossein and Lévesque [6] computed the elastic properties of composites reinforced by spherical particles. Khan et al. [7] generated VEs and RVEs of similar microstructures and used a finite element (FE) method in order to analyze coupled heat conduction and thermal stress variations. Several analytical micromechanical models have been derived for linearly elastic composites, like dilute solution [8], Mori–Tanaka [9,10] (MT), differential [11] and self-consistent [12] (SC) models. Dunn and Taya [13] used these models to evaluate the effective properties of piezoelectric composites with linearly electromechanical constitutive relations. Nan [14] proposed an

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effective-medium theory to predict the effective electro-mechanical properties of piezoelectric composites. Aboudi [15] considered linear electro-magneto-mechanical coupling constitutive relations and used the method of cells, in which the composite microstructures were idealized with periodically distributed arrays of cubic particles. Kim [16] presented an exact matrix method to predict the effective properties of magneto-electro-thermo-elastic multilayer composites and the results were identical to those evaluated from the MT model. Polarized piezoelectric materials can exhibit significantly nonlinear responses when subjected to large electric fields and/or stresses, which is often the case in actuator applications. Tiersten [17] proposed nonlinear constitutive relations of polarized ferroelectric ceramics by considering higher-order terms of the electric field and was able to describe the nonlinear electro-mechanical coupling behavior of lead zirconate titanate (PZT). Currently available micromechanics models with electro-mechanical coupling effects have been primarily focused on linearly electro-mechanical constitutive relations. Only limited micromechanical models considered the nonlinear responses of polarized piezoelectric composites (Tan and Tong [18], Muliana and Lin [19]). In this work, we modified the micromechanical models presented in Khan et al. [7] to study the effective electro-mechanical properties of active composites. We derived effective electromechanical properties for the composite with active particles and compared these properties with those derived from the MT, SC and simplified unit-cell (UC) micromechanical models. Next, we dealt with the nonlinear electro-mechanical response due to large applied electric fields and small deformation gradients and we compared the results from MT and UC models with those obtained from RVEs. The manuscript is organized as follows. Section 2 discusses the nonlinear electro-mechanical coupling model together with nonlinear extensions of MT and UC models. Section 3 presents the numerical procedure used to derive the reference predictions. Section 4 presents the comparison between the predictions of the numerical and analytical homogenization models. Finally, Section 5 summarizes the outcomes of this work. 2. Overview of constitutive models and simplified micromechanical models for nonlinear piezoelectric composites

Di ¼ eikl ekl þ jeij Ej þ

1 e v Ej Ek : 2 ijk

ð2bÞ

The strains and electric fields are considered as independent field variables and the material properties are the elastic stiffness C Eijkl at a constant reference electric field, the third and fourth order piezoelectric stress coefficients eijk and bijkl, and the second and third order dielectric coefficients jeij and veijk at a constant reference strain. The material properties of Eqs. (1) and (2) are related through

sEijkl ¼ C E1 ijkl ;

ð3aÞ

dijk ¼ eimn sEjkmn ;

ð3bÞ

jrij ¼ jeij þ eimn djmn ;

ð3cÞ

fijkl ¼ bijmn sEklmn ;

ð3dÞ

vrijk ¼ veijk þ eimn fjkmn :

ð3eÞ

Eq. (2) can be cast under a secant linearized form as:

R ¼ LðEÞ : Z;

ð4aÞ

where RT ¼ fr11 ; Z T ¼ fe11 ;

r22 ; r33 ; r23 ; r13 ; r12 ; D1 ; D2 ; D3 g;

ð4bÞ

e22 ; e33 ; 2e23 ; 2e13 ; 2e12 ; E1 ; E2 ; E3 g;

2

C E1111 6 E 6 C 1122 6 6 E 6 C 1133 6 6 0 6 6 LðEÞ ¼ 6 0 6 6 6 0 6 6 0 6 6 4 0 e311

ð4cÞ

C E1122 C E1133

0

0

0

0

0

C E2222 C E2233

0

0

0

0

0

C E2233 C E3333

0

0

0

0

0

0

0

C E2323

0

0

0

h223

0

0

0

C E1313

0

h123

0

0 e113 0 0

C E1212

0 g 11 0 0

0 0 g 22 0

0 0 0

0 0 0

e322

e333

0 0 e223 0

0 0 0

h311

3

7 h322 7 7 7 h333 7 7 0 7 7 7 : 0 7 7 7 7 0 7 0 7 7 7 0 5

ð4dÞ

g 33

In Eq. (4d) the nonlinear components of hkij and gij are

2.1. Constitutive models

1 hkij ¼ ekij þ bklij El ; 2

This study focuses on the nonlinear constitutive relations for a polarized piezoelectric material undergoing large electric fields and small strains, proposed by Tiersten [17] as

g ij ¼ jeij þ

1 e v Ek : 2 ijk

ð4eÞ

ð4fÞ

eij ¼ sEijkl rkl þ dkij Ek þ fklij Ek El ;

1 2

ð1aÞ

2.2. Homogenization scheme

1 r v Ej Ek ; 2 ijk

ð1bÞ

For a piezoelectric particle-reinforced composite, the generalized effective field variables are obtained by volume average as:

Di ¼ dikl rkl þ jrij Ej þ

where i, j, k, l = 1, 2, 3. The field variables in these two equations are the stresses rij, strains eij, electric fields Ei and electric displacements Di. The material parameters are the elastic compliances sEijkl at a constant reference electric field, the third and fourth order piezoelectric strain coefficients dijk and fijkl, and the second and third order dielectric coefficients jrij and vrijk at a constant reference stress. Tiersten [17] provides another form of the constitutive relation as

1 2

rij ¼ C Eijkl ekl  ekij Ek  bklij Ek El ;

ð2aÞ

Z¼

X

cr Z r ;

ð5aÞ

cr Rr ;

ð5bÞ

r¼m;p

R¼

X

r¼m;p

where the Zr and Rr are the spatially averaged field variables for each constituent r, the subscripts m and p denote the matrix and the particle phase, respectively, and cr is the volume fraction (vf) of constituent r that satisfies:

X r¼m;p

cr ¼ 1:

ð6Þ

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PZT G-1195 (particle)

Araldit D (matrix)

C E1111 C E1122 C E1133 C E2222 C E3333 C E2323

(GPa)

63

8.0

(GPa)

34

4.4

(GPa)

31

4.4

(GPa)

63

8.0

(GPa)

49

8.0

(GPa) d311 (nm/V) d333 (nm/V) d113 (nm/V) jr11 (109 F/m) jr33 (109 F/m) b1113 (105 F/m) b2223 (105 F/m) b3333 (105 F/m) b3311 (105 F/m) b3322 (105 F/m)

22

1.8

0.18 0.36 0.54 15.1 15.1 3.96 3.96 2.16 2.1 2.1

0.0 0.0 0.0 0.035 0.035 0.0 0.0 0.0 0.0 0.0

Up to this point, the localization tensor is the only unknown. Several micromechanical models have been devoted to determine the localization tensor.

Fig. 1. Unit-cell model with 8 subcells.

~ for the composite The generalized effective stiffness tensor L meets the following equality:

~ : Z: R¼L

ð7Þ

The averaged local strains and electric fields are classically expressed as

2.2.1. MT model The MT model is based on Eshelby’s [8] solution for an ellipsoidal inclusion embedded in an infinite elastic matrix. The details of the MT model can be found in [9,10] and [20]. The localization tensor for the MT model (with r = 0 denoting matrix constituent and r = 1 particle) is 1

dil AMT ¼ Adil r r : ½c 0 I þ c1 A1  ;

ð11aÞ 1

Z r ¼ Ar : Z;

ð8Þ

where Ar is the localization tensor of constituent r and satisfies

X

cr Ar ¼ I;

ð9Þ

r¼m;p

where I is a 9  9 matrix where Iii = 1and 0 otherwise. Finally, the effective generalized stiffness tensor of the active composite can be derived from Eqs. (5)–(8) as

~¼ L

X r¼m;p

cr Lr : Ar :

ð10Þ

1 where Adil r ¼ ½I þ S r : L0 : ðLr  L0 Þ :

ð11bÞ

In Eq. (11b), Sr is the Eshelby tensor of phase r, which had been extended by Deeg [21] and reported by Dunn and Taya [22] for an ellipsoidal inclusion embedded in an infinite linearly piezoelectric medium. 2.2.2. SC model The SC model was derived through the extension of the dilute model and was detailed by Hill [12]. The localization tensor for the SC model (with r = 0 denoting matrix constituent and r = 1 particle) is

Fig. 2. Example of FE meshes of a VE model with randomly distributed solid spherical particles and its components: (a) VE model, (b) matrix, and (c) particles.

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Fig. 3. Average of data with 95% confidence interval for vf = 20%.

Fig. 4. Average of data with 95% confidence interval for vf = 30%.

~ ~1 : ðLr  LÞ ~ 1 : ASC r ¼ ½I þ S r : L

ð12Þ

In Eq. (12), a tilde on the Eshelby tensor indicates that its com~ This ponents should be evaluated with the effective stiffness L. model yields an implicit algebraic 9  9 matrix equation for evalu~ Solutions to this ating the effective generalized stiffness tensor L. implicit matrix equation require numerical iterations. 2.2.3. UC model The UC model of a particle-reinforced composite is approximated as an ideal composite with periodic and uniform distribution of cubic elements. A unit cell has eight subcells and is considered as a simplified microstructure of the composite, as shown in Fig. 1. The interfaces between all subcells are assumed perfectly bonded. The first subcell represents the particle constituent (r = 1) and the rest of the subcells represents the matrix constituent (r = 2, 3, . . ., 8). The field variables in each subcell are assumed as spatially constant. Using the subcell’s constitutive

relations together with linearized micromechanical relations between subcells (see Lin and Muliana [23]), the relationship between the independent field variables of the unit cell and its subcells is

8 9 Z1 > > > > > >Z > > < 2= ¼ ½G1 ½HfZg; . .. > > > > > > > : > ; Z8

ð13Þ

where G is a function of the linearized material parameters and the volume fraction of each subcell and H is a constant matrix. From Eq. (13), the localization matrix becomes

3 AUC 1 6 UC 7 6 A2 7 7 6 1 6 . 7 ¼ ½G ½H: 6. 7 4. 5 2

AUC 8

ð14Þ

V. Tajeddini et al. / Computational Materials Science 82 (2014) 405–414

409

Fig. 5. Mechanical properties from  unit-cell, D self-consistent, h Mori–Tanaka and  RVE-FE approaches.

3. Numerical RVEs for obtaining the reference data The localization tensor of Eqs. (11), (12) and (14) is applicable for the linear constitutive relations. One way to deal with the homogenization of nonlinear constitutive relations is to introduce a linear comparison material having the exact same morphology but for which the linearized properties are attributed to nonlinear phases. In this work, we adopted a secant linearization, as described in Eq. (4), with the reference level being the average value of Z, similarly to what had been done by Berveiller and Zaoui [24], so that:

Rr ¼ Lr ðEr Þ : Z r :

ð15Þ

This linearization procedure leads to an implicit problem: Ar depends on the linearized properties, Zr, and at the same time Zr depends on Ar. This implicit problem was solved with a fixed-point method. The convergence was achieved when the ðnþ1Þ norm of ðZ ðnÞ Þ was smaller than the threshold value r  Zr 6 10 . In this study, 20 iterations were typically required to reach convergence. It is obvious from Fig. 1 that the effective properties predicted from the UC model are of cubic symmetry and that the model is tailored for regular arrangement of cubic particles. However, all other models (both analytical and numerical) represent (approximately for the numerical models) isotropic distributions of spherical particles. The UC models are therefore approximations of the materials studied in this work.

The studied active composites consisted of spherical particles distributed randomly in a homogeneous matrix. Volume elements (VEs) with imposed periodic boundary conditions were meshed with 10-node tetrahedron elements and implemented in ABAQUS FE (Fig. 2). The reader can refer to Barello and Lévesque [25] for more details on the VE generation. Two active composite systems were considered: polarized PZT spherical particles distributed in an elastic matrix and a polarized PZT matrix with elastic spherical particles. We also considered VEs of a polarized PZT matrix with porosity. The following PZT and elastic constituents were used: PZT G-1195 and Araldit D polymer. The mechanical and electrical properties of these constituents were obtained from [23] and are listed in Table 1. It should be noted that the nonlinear permittivity coefficients were taken as zero due to lack of available data for PZT G-1195. The piezoelectric components, d322, d223 and jr22 were set equal to d311, d113 and jr11 , respectively. VEs with two inclusion volume fractions of 20% and 30% were considered. We followed the methodology proposed by Kanit et al. [26] for obtaining the overall responses. To determine the representative volume element (RVE) for a fixed volume fraction and piezoelectric properties, we derived the desired effective response for a specific number of particles in the matrix. Then, we increased the number of particles and computed the effective response. We continued increasing the number of particles until no significant changes in the predicted effective properties were obtained. For each volume fraction and specific number of particles, we analyzed several realizations (e.g. models

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Fig. 6. Electrical properties from  unit-cell, D self-consistent, h Mori–Tanaka and  RVE-FE approaches.

Fig. 7. Strain responses of the active composite incorporating nonlinear electric field effect under normal stress with left: 20% and right: 30% of particles. The error bars represent a 95% confidence interval on the mean value.

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411

Fig. 8. Strain responses of the active composite incorporating nonlinear electric field effect under electric field with left: 20% and right: 30% of particles. The error bars represent a 95% confidence interval on the mean value.

Fig. 9. Strain responses of the active PZT-matrix composite of particles incorporating nonlinear electric field effect under normal stress with left: 20% and right: 30% of particles. The error bars represent a 95% confidence interval on the mean value.

Fig. 10. Strain responses of the active PZT-matrix composite with vf = 30% of particles incorporating nonlinear electric field effect under electric field with left: 20% and right: 30% of particles. The error bars represent a 95% confidence interval on the mean value.

with different arrangement of particles) and calculated the effective properties. One of the advantages of the considered RVE-FE models is that they allow for capturing the concentration/localized field variables, such as stress discontinuities, and electric field concentration. High values of localized field variables in reality could induce damages, such as cracking, and debonding between the particle and matrix. When the existence of localized field variables is considered significant, there is a limitation in the magnitude of the prescribed electric field and stress that could be applied in order to avoid potential damage. We did not consider any debonding threshold and/or damage criterion in our simulation. However, high localized field variables could lead to very slow convergence and even divergence in the solution especially for the cases with nonlinear electric field effect and larger volume fraction of active elements. The maximum net electric field prescribed to the composites was 1 MV/m for the first composite system and 0.1 MV/m for the second composite system. The maximum net normal stress prescribed was 1 MPa for both of the systems.

4. Results and discussions 4.1. Effective linear electro-mechanical properties Figs. 3 and 4 show the average and 95% confidence intervals for some of the electro-mechanical properties, elastic axial modulus Y3, shear modulus G23 and piezoelectric constants d223 and d113 for the composites with vf = 20% and 30% of active particles, respectively (the plots shown are the best and the worst three properties in terms of their scatter). Similarly, some of the electro-mechanical properties are shown in Fig. 4. Four realizations were computed for each configuration. For vf = 20%, the largest relative width of the confidence interval was for d223 at 9.5% for the VE with 20-particles. It was less than 6% for all the other properties. For composites with vf = 30%, the largest width of the confidence intervals were obtained for shear related piezoelectric coefficients, namely: 16.5%, 8% and 7% for 15, 30 and 45 particles for d223 and 15.5%, 7.5% and 6.5% for 15, 30 and 45 particles for d113. It was less than 7% for all the other properties. Increasing

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Fig. 11. Linear (L) and nonlinear (NL) strain responses of active piezo-ceramic with vf = 20% and 30% porosity. The error bars represent a 95% confidence interval on the mean value.

Fig. 12. Strain responses of the active piezo-ceramic with vf = 20% porosity incorporating left: only linear electric field effect and right: nonlinear electric field effect.

Fig. 13. Strain responses of the active piezo-ceramic with vf = 30% porosity incorporating Left: linear electric field effect and right: nonlinear electric field effect.

the number of repetitions, as was done in Ghossein and Lévesque [6] could have reduced the uncertainty in these average properties. However, for this study, these relative widths of the confidence intervals were judged as acceptable and were used for statistical comparison. Based on these results, it can be seen that the RVE was reached for all properties, except for d223 for the composites containing 20% of reinforcements. As a result, we chose 40 particles for vf = 20% and 45 particles for vf = 30% as our RVEs. The predictions of these models are labeled as RVE-FE herein. The overall linear mechanical properties of the active composites having PZT G-1195 particles distributed in Araldite matrix are shown in Fig. 5. The polarization axis of the PZT particles was in the x3 direction. The responses were reported for the composites with vf = 20% and 30% of particles obtained from the RVE-FE, MT, UC and SC models. We used vn to denote the VEs with n number of particles, e.g., v20 refers to a VE with 20 particles. For the elastic axial moduli, the UC model generally gives the closest prediction to the RVE-FE models. For the shear modulus, larger differences are seen for the UC and MT models while

the SC predictions are relatively close to those of the RVE-FE models. Fig. 6 shows the overall piezoelectric and permittivity constants, derived from the RVE-FE, MT, UC, and SC models. It is seen that for the piezoelectric strain coefficients, the UC and MT results are in good agreement with those predicted by RVE-FE models. The SC solution presents large discrepancies, as it is also the case for the elastic constants obtained from the SC model. 4.2. Nonlinear response of active composites For the nonlinear analyses, we implemented the nonlinear electro-mechanical constitutive models (Eqs. (1) and (2)) in a user element (UEL) subroutine of ABAQUS. Two active composite systems were considered. The first system consisted of PZT particles distributed in an elastic matrix, while the second system considered elastic particles randomly distributed in a piezoelectric matrix. Two types of loadings were studied for each system: (1) a uniform axial tensile stress was applied along the poling axis (x3 direction) and

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413

Fig. 14. Strain responses of RVE-FE model of active piezo-ceramic with different volume fractions of porosity and boundary conditions.

the two faces normal to that axis were kept under zero electric potential condition while maintaining traction free conditions on the other transverse directions; (2) a potential difference Du was applied on the two surfaces with poling axis as normal direction so that the electric field was along the positive poling direction and traction free conditions were imposed on the surfaces of the active composites. The net electric field in the poling direction was determined by E3 ¼ DDx/3 . Axial strains e33 were determined for both loadings. Fig. 7 shows the strain response in direction of poling axis under an applied electric field and stress in the same direction for the composite with vf = 20% of piezoelectric particles (composite systems 1). The deformation responses from FE model for 20 and 40 represented particles are fairly close to each other. 95% confidence intervals were also computed for specific values of the applied load. Fig. 7 shows that the two curves are not statistically different and the RVE was assumed to be constituted of 40 particles. Fig. 8 shows the corresponding results for the composites with vf = 30% of the particles. It can be seen that the RVE was not reached for the number of VE simulated under applied electric field. However, the fact that the predictions for 45 represented particles lies between those for 15 and 30 particles suggests that performing more realizations should lead to the RVE. We therefore considered 45 particles as an approximate RVE. In Figs. 7 and 8 we observe that when the composite is subjected to applied stresses, the MT model shows a good agreement with the RVE-FE model, i.e., the differences with the RVE-FE solution are in the range of 2–7% but are 25–28% for the UC solutions. When the composite is subjected to electric fields, the differences of results from the RVE-FE solutions are in the range of 15–26% for the MT model and 14–21% for the UC solutions. Thus, the accuracy of MT and UC are almost the same for this case. It can also be seen that the overall strain response due to the variation of electric field is somewhat linear despite of the high magnitude of applied

electric field. This could be due to the low volume contents of the piezoelectric particles and to the fact that the non-active matrix dominates the overall behavior of the composites. Figs. 9 and 10 illustrate the strain responses of the composites with a piezoelectric matrix and elastic particles. 4 Realizations were analyzed for each response and very narrow confidence intervals were obtained. These figures clearly show the nonlinear response since 70% and 80% of the composite consists of a piezoelectric constituent. The responses were obtained from the MT, UC and RVE-FE models. The figures also show that the RVE was reached in all simulated cases. In overall, good agreements between the RVE-FE and UC and MT models were obtained for the response under the applied stress. The MT model predicts the deformation responses better than UC; the maximum differences of UC results from those of the RVE-FE models is about 13% while for the MT model, the differences are less than 2%. On the other hand, under applied electric field, the MT model overestimates the response while the UC model delivers closer predictions; the maximum discrepancy in the predictions of the MT and UC models with respect to those of the RVE-FE models were of 37% and 17%, respectively. These important discrepancies might be attributed to the secant linearization procedure used, known to deliver poor estimates [27] in the case on nonlinear mechanical responses. Furthermore, from these two figures it is seen that while the deformation increases linearly due to increase of the applied stress (at least up to 1 MPa normal stress), it increases nonlinearly with decreasing rate due to increase of the applied electric field. 4.3. Response of porous piezoelectric materials The RVE-FE models for the particulate composites were modified by eliminating the particles in the matrix in order to model the piezo-ceramics with porosity. The studied piezoelectric ceramics were with 20% and 30% porosity and with 40 and 45 pores

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respectively for RVE-FE models. Fig. 2(b) illustrates the porous piezo-ceramics model with vf = 20% of pores. Fig. 11 shows the linear and nonlinear strain-electric field responses for the piezoceramics with 20% and 30% porosity. The piezo-ceramics were subjected to an electric field applied along the poling direction, direction 3, and under a stress-free condition. Here again, an overlap of the responses from VEs with different number of pores and very small 95% confidence intervals shows that the RVEs were reached. It is seen that under a small electric field the effect of nonlinear electric field term is almost negligible. The effect of electric field nonlinearity on the response becomes significant when the composites are under larger electric fields. Next, the results of RVE-FE model were used to evaluate MT and UC prediction. Figs. 12 and 13 show the results for piezo-ceramics with vf = 20% and 30% of pores. It can be seen that for linear constituents the MT model predicts the responses with much better accuracy than the UC model. However, the UC model delivers much more accurate predictions when nonlinear properties are considered. Finally, the effect of volume fraction of pores on the overall responses of the composites under different types of boundary conditions and loads is shown in Fig. 14. From Fig. 14(a), it is seen that effect of electric field input on the deformation response is not affected by the volume fraction of pores. This is because the disturbance in the electric potential distribution through the medium due to the pores is small enough and so the magnitude of the electric field in the medium does not change significantly; therefore, by considering Eqs. (1a) and (1b), under free stress and electric displacement conditions, the strain remains nearly the same. However, when the inputs loads such as force or electric charge are considered, the existence of porosity increases the overall deformation, as seen in Fig. 14(b and c). Higher porosity decreases the macroscopic mechanical resistance to the deformations, or reduces the overall elastic stiffness of the materials. 5. Summary In this study, we compared the overall (macroscopic) electromechanical response of piezoelectric composites determined from micromechanical models with simplified microstructural morphologies, i.e., MT, UC, and SC, to those obtained from RVE-FE results. We considered the RVE-FE results as our numerical reference data since they consist of more realistic microstructural morphologies and therefore are capable of capturing variations in field variables (stress, strain, electric field, and electric flux) within their microstructures. It was concluded that the MT and UC models are capable of predicting the overall linear properties of the active composites for the range of parameters studied in this work. Although they are in relatively good agreement with RVE-FE results for mechanical properties, the SC predictions diverge significantly from the RVE-FE simulations. In the second study, we considered the nonlinear electromechanical constitutive relation. For the composite, comprising of piezoelectric inclusions dispersed in an elastic matrix, it was seen that compared to the UC model, the MT model results are in good agreement with RVE-FE ones when stress is applied but under applied electric field, some deviations from RVE-FE results

were seen for both the UC and MT models. For the studied composite, having elastic inclusions dispersed in piezoelectric matrix, the results of RVE-FE and MT models were almost identical for case of applied stress, while for the case of applied electric field, the UC models had better prediction of results compared to the MT model. Furthermore, it was seen that when the volume fraction of piezoelectric constituent is large enough and the composites are subjected to large electric fields, the effect of electric field nonlinearity on the response becomes important and observable. A direct relation between the volume fraction of the piezoelectric constituent and magnitude of an induced strain under an applied electric field was seen from those two studies, but under an applied stress the inverse relation was seen which was due to larger stiffness of the piezoelectric constituent. Finally, from studying porous piezo-ceramic, it was seen that the porosity has almost no effect on the deformation response when the porous piezo-ceramics are subjected to electric field boundary and traction free conditions. On the other hand, the effect of porosity becomes apparent when the porous piezo-ceramics are under applied stress and/or electric charge. Acknowledgement This research is sponsored by the Air Force Office of Scientific Research (AFOSR) under Grant FA 9550-10-1-0002. We also thank the Texas A&M Supercomputing Facility (http://sc.tamu.edu/) for providing computing resources useful in conducting the research reported in this paper. References [1] Sh Pyun, Y.Y. Jin, G.S. Lee, J. Mater. Sci. Lett. 21 (3) (2002) 243–244. [2] M. Hori, T. Aoki, Y. Ohira, S. Yano, Composites: Part A 32 (2) (2001) 287– 290. [3] A.A. Gusev, J. Mech. Phys. Solids 45 (1997) 1449–1459. [4] J.C. Michel, H. Moulinec, P. Suquet, Comput. Methods Appl. Mech. Eng. 172 (1999) 109–143. [5] J. Segurado, Llorca, J. Mech. Phys. Solids 50 (2002) 2107–2121. [6] E. Ghossein, M. Levesque, Int. J. Solids Str. 49 (11 and 12) (2012) 1387–1398. [7] K.A. Khan, R. Barello, A.H. Muliana, M. Levesque, Mech. Mater. 43 (2011) 608– 625. [8] J.D. Eshelby, Proc. Roy. Soc. London Ser. A 252 (1957) 561–569. [9] T. Mori, K. Tanaka, Acta Metall. Mater. 21 (1973) 571–574. [10] Y. Benvensite, Mech. Mater. 6 (2) (1987) 147–157. [11] R. McLaughlin, Int. J. Eng. Sci. 15 (4) (1977) 237–244. [12] R. Hill, J. Mech. Phys. Solids 13 (1965) 213–222. [13] M.L. Dunn, M. Taya, Int. J. Solids Struct. 30 (2) (1993) 161–175. [14] C.-W. Nan, J. Appl. Phys. 76 (2) (1994) 1155–1163. [15] J. Aboudi, Smart Mater. Struct. 10 (5) (2001) 867–877. [16] J.-Y. Kim, Int. J. Eng. Sci. 49 (9) (2011) 1001–1018. [17] H.F. Tiersten, J. Appl. Phys. 74 (5) (1993) 3389–3393. [18] P. Tan, L. Tong, Compos. Sci. Technol. 61 (5) (2001) 759–769. [19] A. Muliana, C.-H. Lin, J. Intel. Mat. Syst. Str. 22 (8) (2011) 723–738. [20] G.J. Weng, Int. J. Eng. Sci. 22 (7) (1984) 845–856. [21] W.F. Deeg, The Analysis of Dislocation, Cracks, and Inclusion Problems in Piezoelectric Solids. Ph.D. Dissertion, Stanford University, United States of America, 1980. [22] M.L. Dunn, M. Taya, Proc. R. Soc. London A 443 (1993) 265–287. [23] C.-H. Lin, A. Muliana, Acta Mech. 224 (7) (2013) 1471–1492. [24] M. Berveiller, A. Zaoui, J. Mech. Phys. Solids 26 (5–6) (1979) 325–344. [25] R.B. Barello, M. Lévesque, Int. J. Solids Struct. 45 (2008) 850–867. [26] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Int. J. Solids Str. 40 (2003) 3647–3679. [27] A. Rekik, M. Bornert, A. Zaoui, F. Auslender, Evaluation of Linearization Procedures Sustaining Nonlinear Homogenization Theories, Proc. XXI ICTAM, Poland, 2004.