Constitutive Modeling of Piezoelectric Polymer Composites

Transcription

1 Acta Materialia, l. 5, n.18, pp (004) Cnstitutive Mdeling f Piezelectric Plymer Cmpsites G.M. Odegard Michigan Technlgical niversity Department f Mechanical Engineering Engineering Mechanics 1400 Twnsend Drive Hughtn, MI Abstract A new mdeling apprach is prpsed fr predicting the bulk electrmechanical prperties f piezelectric cmpsites. The prpsed mdel ffers the same level f cnvenience as the wellknwn Mri-Tanaka methd. The electrmechanical prperties f fur piezelectric plymer cmpsite materials are predicted with the prpsed, Mri-Tanaka, Self-cnsistent methds, and detailed finite element analyses are cnducted ver full ranges f reinfrcement vlume fractins. The presented data ffer a cmprehensive cmparisn f the fur mdeling appraches fr a wide range f matrix and reinfrcement electrmechanical prperties, reinfrcement gemetry, and reinfrcement vlume fractin. By cmparisn with the finite element data, it is shwn that the prpsed mdel predicts prperties that are, in sme cases, mre accurate than the Mri-Tanaka and Self-cnsistent schemes. Keywrds: Micrmechanics; Mri-Tanaka; Piezelectricity; PZT; PDF; Self-cnsistent Intrductin Piezelectric materials are excellent candidates fr use in sensrs and actuatrs because f their ability t cuple electrical and mechanical energy. Fr sme applicatins, it is necessary t use cmpsite materials in which ne r mre f the cnstituents have piezelectric prperties. T facilitate the design f these piezelectric cmpsite systems, cnvenient and accurate structureprperty relatinships must be develped. Numerus attempts have been made t develp mdels t relate bulk electrmechanical prperties f cmpsite materials t the electrmechanical prperties f individual cnstituents. Simple estimates, utilizing igt r Reuss-type appraches, have been used t predict the behavir f a limited class f cmpsite gemetries [1-4]. pper and lwer bunds fr the electrmechanical mduli have been determined [5-8]. Finite element analysis has als been used t predict electrmechanical prperties [9, 10]. Even thugh finite element analysis has the best ptential fr accurately predicting cmpsite prperties fr any cmpsite gemetry, the slutins can be very expensive and time-cnsuming. Several authrs have extended Eshelby s [11] classical slutin f an infinite medium cntaining a single ellipsidal inclusin t include piezelectric cnstituents [1-15]. Als referred t as the dilute slutin, this apprach ignres the interactins f the inclusins that ccur at finite inclusin vlume fractins. Other studies [14, 16-19] have fcused n the classical extensins f Eshelby s slutin fr finite inclusin vlume fractins, i.e., the Mri-Tanaka [0, 1], Self-

2 cnsistent [, 3], and differential [4, 5] appraches. Analytical slutins fr specific cmpsite systems have als been determined [6-3]. Even thugh the verall framewrk f these appraches prvides estimates fr a wide range f inclusin sizes, gemetries, and rientatins, each f these methds suffers frm drawbacks assciated with accuracy and cmputatinal cnvenience. In this paper, a mdel is prpsed fr predicting the cupled electrmechanical prperties f piezelectric cmpsites. This mdel is an extensin f a technique riginally develped fr predicting mechanical prperties f cmpsites by generalizing the Mri-Tanaka and Selfcnsistent appraches [33]. First, the verall cnstitutive mdeling f piezelectric materials is discussed, fllwed by a descriptin f the prpsed mdel. Finally, the electrmechanical prperties f fur different piezelectric cmpsite systems are predicted using the prpsed, Mri-Tanaka, Self-cnsistent, and finite element mdels. The fur piezelectric cmpsite systems used in this study were chsen t represent a wide range f practical materials: a graphite/ply(vinylidene fluride) (PDF) cmpsite, a Silicn Carbide (SiC)/PDF particulate cmpsite, a fibrus Lead Zircnate Titanate (PZT)/plyimide cmpsite, and a PZT/plyimide particulate cmpsite. Cnstituent materials The matrix and inclusin cnstituents used in this study were chsen such that the cmpsite materials had fur cmbinatins f piezelectric cnstituents and reinfrcement gemetries. The graphite/pdf and SiC/PDF cmpsites have a piezelectric plymer matrix with fiber and particle reinfrcement, respectively. The PZT/plyimide cmpsites have a piezelectric inclusin with fiber and particle reinfrcements. PDF is a rthtrpic, semi-crystalline plymer which exhibits a piezelectric effect with an electric field applied alng the 3-axis. Typical electrmechanical prperties f PDF are given in Table 1 (these prperties were supplied by NASA Langley Research Center). LaRC-SI is a thermplastic plyimide that was develped fr aerspace applicatins. The prperties f LaRC- SI used in this study crrespnd t the system with a 3% stichimetric imbalance at rm temperature [34] and are als shwn in Table 1. The PDF plymer was reinfrced with bth infinitely-lng graphite fibers and spherical SiC particles. The fibers were unidirectinally aligned alng the PDF 1-axis. This alignment was chsen fr the mdeling because f the desire t maintain a high level f material cmpliance (therefre maximizing the piezelectric effect) in the transverse directins, while prviding reinfrcement in the directin in which little piezelectric effect and maximum mechanical reinfrcement are required. The LaRC-SI plymer was reinfrced with bth infinitely-lng PZT-7A fibers and spherical PZT-7A particles. PZT-7A is a ceramic that exhibits a piezelectric effect with electric fields applied alng all three principle axes. The PZT-7A fibers were unidirectinally aligned with the fiber 3-axis as the fiber-length axis. This alignment was chsen t maintain cnsistency with previus analyses [8, 14, 19], which ultimately prvides alignment f the fibers during the pling prcess in the fabricatin f these materials. All f the inclusin electrmechanical prperties are given in Table 1.

3 Micrmechanics mdeling 1. Piezelectric materials There are three standard ntatin systems that are cmmnly used t describe the cnstitutive mdeling f linear-piezelectric materials. sing the cnventinal indicial ntatin in which repeated subscripts are summed ver the range f i,j,m,n = 1,,3, the cnstitutive equatins are σ = C ε + e E ij ijmn mn nij n D = e ε κ E i imn mn in n (1) where σ ij, ε ij, E i, and D i are the stress tensr, strain tensr, electric field vectr, and the electric displacement vectr, respectively. The quantities C ijmn, e nij, and κ in are the elastic stiffness tensr, the piezelectric tensr, and the permittivity tensr, respectively. The divergence equatins, which are the elastic equilibrium and Gauss law, are, respectively, σ ij, j D ii, () where the subscripted cmma dentes partial differentiatin. The gradient equatins, which are the strain-displacement equatins and electric field-ptential, are, respectively, 1 ε = + E = φ ( u, u, ) ij i j j i i, i (3) where u i and φ are the mechanical displacement and electric ptential, respectively. In the mdeling f piezelectric materials, it is mre cnvenient t restate Eqn. (1) s that the elastic and electric variables are cmbined t yield a single cnstitutive equatin. This ntatin is identical t the cnventinal indicial ntatin with the exceptin that lwer case subscripts retain the range f 1-3 and capitalized subscripts take n the range f 1-4, with repeated capitalized subscripts summed ver 1-4. In this ntatin, Eqn. (1) is where Σ ij, E ijmn, and Z Mn are, respectively, Σ ij = EiJMnZ Mn (4) Σ ij σij = D i J =1,,3 J =4 (5) 3

4 E ijmn Cijmn J, M = 1,,3 enij J = 1,,3; M = 4 = eimn J = 4; M = 1,,3 κ in J, M = 4 (6) Z Mn ε mn M = 1,, 3 = En M = 4 (7) The piezelectric cnstitutive equatin can be further simplified by expressing Eqn. (4) in matrix ntatin Σ = ΕZ (8) where the bldface indicates either a 9 9 matrix (E) r a 9 1 clumn vectr (Σ, Z) [ σ σ σ σ σ σ D D D ] t Σ = (9) [ E E E ] t Z = ε11 ε ε33 γ3 γ13 γ1 1 3 (10) t C e ( 6 6) ( 6 3) E = e κ ( 3 6) ( 3 3) (11) In Eqn. (11), C, e, and κ dente the elastic stiffness matrix, the piezelectric cnstant matrix, and the permittivity matrix, respectively. The superscript t dentes a matrix transpsitin. Nte that γ ij = ε ij in rder t keep E a symmetric matrix. Frm Eqns (8) - (11), the cnstitutive equatin fr an rthtrpic piezelectric material is σ11 C11 C1 C e31 ε11 σ C1 C C e 3 ε σ33 C13 C3 C e33 ε33 σ C e15 0 γ3 σ 13 = C55 0 e γ 13 σ C γ1 D e15 0 κ1 0 0 E 1 D e κ 0 E D e e e κ E (1) 4

5 where the cntracted igt ntatin is used. In Eqn. (1), the 3-axis is aligned with the principle directin f plarizatin.. Electrmechanical prperties f cmpsites sing the direct apprach [14, 35, 36] fr the estimate f verall prperties f hetergeneus materials, the vlume-averaged piezelectric fields f the cmpsite with a ttal f N phases are = N Σ crσ r (13) r= 1 = N Ζ crζ r (14) r= 1 where c r is the vlume fractin f phase r, the verbar dentes a vlume-averaged quantity, the subscript r dentes the phase, and r = 1 is the matrix phase. The cnstitutive equatin fr each phase is given by Eqn. (8). Fr a piezelectric cmpsite subjected t hmgeneus elastic strain and electric field bundary cnditins, Z 0 0, it has been shwn that Z= Z [16]. The cnstitutive equatin fr the piezelectric cmpsite can be expressed in terms f the vlumeaveraged fields The vlume-average strain and electric field in phase r is where A r is the cncentratin tensr f phase r, and Σ = EZ (15) Zr = Ar Z (16) N cra r= 1 r = I (17) where I is the identity tensr. Cmbining Eqns. (13)-(17) yields the electrmechanical mdulus f the cmpsite in terms f the cnstituent mduli 1 N r= E= E + c E E A r r 1 r (18) arius prcedures exist fr evaluating the cncentratin tensr. The mst widely used appraches are the Mri-Tanaka and Self-cnsistent schemes. 5

6 Fr the Mri-Tanaka apprach, the cncentratin tensr is 1 N dil dil As = Ar c1i+ cra r (19) r= where A dil r is the dilute cncentratin tensr given by 1 dil 1 A = + r I SrE1 Er E1 (0) In Eqn. (0) S r is the cnstraint tensr fr phase r, which is analgus t the Eshelby tensr used in determining elastic prperties f cmpsite materials [11]. The cnstraint tensr is evaluated r as a functin f the lengths f the principle axes f the reinfrcing phase r, a i, and the electrmechanical prperties f the surrunding matrix ( 1, 1,, 3) S = E r r r r f a a a (1) The cmplete expressin fr Eqn. (1) is given elsewhere [16]. While the Mri-Tanaka apprach prvides fr a quick and simple calculatin f the bulk cmpsite electrmechanical prperties, it has been shwn that it yields predicted mechanical prperties that are relatively lw and high fr cmpsites with stiffer inclusins and matrix, respectively [33]. This issue culd pssibly lead t less accurate estimatins f the electrmechanical mduli, especially fr relatively large inclusin vlume fractins [37-39]. In the Self-cnsistent scheme, the cncentratin tensr is 1 1 A = + r I SrE Er E () where E is the unknwn electrmechanical mduli f the cmpsite, and the cnstraint tensr, r S r, is evaluated as a functin f E and a i. Since the electrmechanical mduli f the cmpsite appears in bth Eqns. () and (18), iterative schemes r numerical techniques are ultimately required fr the predictin f the electrmechanical mduli f cmpsites using the Selfcnsistent methd. This apprach results in slw and cmplicated calculatins. It has been demnstrated [33] that a mre general frm f the cncentratin tensr can be used fr the predictin f mechanical prperties f cmpsites. Extending this cncept t the predictin f electrmechanical prperties results in 1 1 A r = I+ SrE0 Er E0 (3) where E 0 is the electrelastic mduli f the reference medium, and the cnstraint tensr is r evaluated using E 0 and. Therefre, it is assumed that the reference medium is the material a i 6

7 that immediately surrunds the inclusin fr the evaluatin f the cnstraint and cncentratin tensrs. Naturally, the electrelastic mduli f the reference medium can have a wide range f values, hwever, it is mst realistic t assume that they are similar t the mduli f the verall cmpsite, as is the case in the Self-cnsistent methd. Fr cnvenience, a simple, yet accurate, estimatin f the verall electrelastic mduli can be chsen fr the reference medium s that the verall prperties f the piezelectric cmpsite can be calculated using Eqns. (18) and (3). Even thugh a simple and accurate estimatin f the reference medium means that the electrelastic mduli can be calculated withut Eqns. (18) and (3), this framewrk allws fr the cmputatin f the mduli fr varius inclusin sizes, gemetries, and rientatins. The reference medium is apprximated with a set f equatins that are similar t the Halpin-Tsai relatin [40], which is extended here fr multiple inclusins and piezelectric cmpsites E = E 1+ η 0 1 r= ijkl ijkl N 1 N r= r ijkl r η c c r ijkl r (4) where E η = E r 1 r ijkl ijkl ijkl r 1 EiJKl + EiJKl (5) Eqns. (4) and (5) indicate that as c 1 1 and c r 1, E 0 ijkl respectively. E and E 0 1 ijkl ijkl r E ijkl, Eqns. (18) and (3)-(5) were used t calculate the electrmechanical prperties fr the fur cmpsite systems fr inclusin vlume fractins ranging frm 0% t the maximum theretical limits, which are abut 90% and 75% fr fibrus and particulate cmpsites, respectively. The cnstraint tensr in Eqn. (3) was evaluated numerically using Gaussian quadrature [41]. The fibers were mdeled as infinitely lng cylinders and the particles were mdeled as spheres. Perfect bnding between the inclusins and matrix was assumed. Finite element analysis Anther apprach t estimate the electrmechanical prperties f piezelectric cmpsites is finite element analysis f a representative vlume element (RE) f the material. Whereas the methds f the previus sectin prvide relatively quick predictins by assuming that the stress and strain fields inside the inclusins are cnstant, finite element analysis predicts these fields in the inclusin and matrix, and thus, prvides a mre realistic predictin t the verall electrmechanical mduli f the cmpsite. This added accuracy cmes at a price, hwever, since each independent prperty f the piezelectric cmpsite (16 independent parameters are shwn in Eqn. (1)) must be determined by a single finite element analysis. In parametric studies where many cmbinatins f inclusin shape and vlume fractin must be cnsidered, 7

8 the finite element apprach can becme very time-cnsuming and expensive. Therefre, in this study, the finite element results are used t check the accuracy f the mdeling methds discussed in the previus sectin. The finite element mdel was develped and executed using ANSYS 7.0. Representative vlume elements (REs) f fiber- and particulate-reinfrced cmpsites were meshed using 10- nded electrmechanical tetrahedral elements with 40 degrees f freedm, three displacements and an electric ptential at each nde (SOLID98). The fibrus cmpsite RE (Fig. 1) simulated a hexagnal packing arrangement, with a maximum fiber vlume fractin f abut 90%. The particulate cmpsite RE (Fig. ) had hexagnal packing in ne plane with a maximum particle vlume fractin f abut 60%. Fr each finite element analysis, the desired vlume fractin was btained by adjusting the dimensins f the RE while keeping the reinfrcement size cnstant. The prperties f the materials are shwn in Table 1. Additinal reinfrcement and matrix material were cnnected t each f the eight faces f bth the fibrus and particulate REs t frm the full finite element mdels (Figs. 1 and ). Fr hmgeneus applied elastic strains and electric fields, the displacements and vltages n the bundary f the full finite element mdels were, respectively, i u B =ε x j φ = ij Ex i i (6) where B indicates the bundary f the full finite element mdel. A ttal f 16 bundary cnditins were applied t the finite element mdels fr each cmbinatin f material type and vlume fractin. Each bundary cnditin was used t predict ne f the independent electrelastic cnstants in Eqn. (1). The electrelastic cnstants and the crrespnding applied strains, electric fields, and the bundary cnditins calculated using Eqn. (6) are listed in Tables t 6. Fr each set f bundary cnditins, all unspecified strains and electric fields in Tables t 6 are zer. It is nted at this pint that the bundary cnditins specified in Eqn. (6) are ften referred t as kinematic bundary cnditins. These bundary cnditins are nt applied directly t the bundary f the RE. Instead, they are applied t the bundary f the full finite element mdel. Therefre, the resulting defrmatins f the RE are nt ver-cnstrained. Over-cnstrained RE edges are a result f applying the kinematic bundary cnditins directly t the bundary f the RE [4]. The elastic strain energy, dielectric energy, and electrmechanical energy f a piezelectric material are, respectively, C n m e = e = ijklεijεkl m= 1 E E n m d = d = κij i m= 1 e j n m em = em = ijkε jkei m= 1 (7) 8

9 m where is the energy f element m, n is the ttal number f finite elements in the RE, and is the vlume f the RE. The energies where calculated fr each element in the RE vlumes fr each set f bundary cnditins (Tables t 6) applied t the full finite element mdel bundary. The ttal energies f the REs were determined by summing the energies f each RE element, as indicated by the first equality in Eqn. (7). The crrespnding elastic, dielectric, and piezelectric cnstants were subsequently calculated using the secnd equality in Eqn. (7). Results and Discussin The Yung s mduli, Y 1, Y, and Y 3 ; shear mduli, G 3, G 13, and G 1 ; piezelectric cnstants, e 15, e 31, e 3, e 33 ; and dielectric cnstants, κ 1 /κ 0, κ /κ 0, and κ 3 /κ 0 ; fr the fur materials discussed in this paper are presented belw. The subscripts f these quantities indicate the crrespnding axes, as shwn in Eqn. (1), and the permittivity f free space, κ 0, is C/m. 1. Graphite/PDF fiber cmpsite The Yung s mduli f the graphite/pdf cmpsite are shwn in Fig. 3 as a functin f the graphite fiber vlume fractin fr the results btained with the finite element analysis, the prpsed mdel discussed abve, the Mri-Tanaka mdel, and the Self-cnsistent methd. Fr the Yung s mdulus parallel t the fiber-alignment directin, Y 1, all fur mdels predict the same values fr the entire range f fiber vlume fractins. Fr the tw transverse mduli, Y and Y 3, the Mri-Tanaka and finite element mdels match very well fr the entire range f fiber vlume fractins, while the prpsed and Self-cnsistent mdels ver-predict the Yung s mduli fr fiber vlume fractins ver 40%. The shear mduli f this material fr the entire range f fiber vlume fractins are shwn in Fig. 4. Fr the lngitudinal shear mduli, G 13 and G 1, the prpsed mdel has a clser agreement with the finite element mdel than the Mri-Tanaka and Self-cnsistent mdels have with the finite element mdel fr fiber vlume fractins abve 40%. Fr the transverse shear mdulus, G 3, the prpsed, Mri-Tanaka, and the Self-cnsistent all predict slight higher values than the finite element mdel, with the Mri-Tanaka shwing the clsest agreement. The piezelectric cnstants, e 31, e 3, and e 33, are shwn in Fig. 5 as a functin f the fiber vlume fractin. The fur mdels predict nearly equal values f e 31 and e 3 ver the entire range. Fr the piezelectric cnstant e 33, the prpsed and Self-cnsistent results ver-predict the finite element mdel, while the Mri-Tanaka methd shws gd agreement with the finite element mdel. The dielectric cnstants, κ 1 /κ 0, κ /κ 0, and κ 3 /κ 0, are shwn in Fig. 6. All fur mdels predict identical values fr all three dielectric cnstants fr the cmplete range f fiber vlume fractins. 9

10 . SiC/PDF particle cmpsite The Yung s mduli f the SiC/PDF cmpsite are shwn in Fig.7 as a functin f particle vlume fractin. At particle vlume fractins f abut 0% and lwer, all fur mdels predict nearly identical mduli. At higher particle vlume fractins, the prpsed mdel predicts mduli that have clser agreement with the finite element results than has the predicted values frm the Mri-Tanaka mdel. Fr particle vlume fractins higher than 0%, the Self-cnsistent apprach significantly ver-predicts the ther three mdels. The three shear mduli are shwn in Fig. 8 fr the entire range f particle vlume fractins. Fr all three shear mduli, at vlume fractins f 50% and less, the Mri-Tanaka and finite element mdels have clse agreement, with the prpsed mdel ver-predicting the shear mduli. Fr a vlume fractin f 60%, the shear mduli f the finite element mdel start increasing dramatically, and the prpsed mdel shws clser agreement with the finite element mdel than des the Mri-Tanaka apprach. Fr particle vlume fractins ver 0%, the Self-cnsistent apprach significantly ver-estimates all three shear mduli. The piezelectric cnstants are shwn in Fig. 9 as a functin f particle vlume fractin. Fr the cnstant e 31, the prpsed mdel data matches the finite element data mre clsely than des the Mri-Tanaka and Self-cnsistent appraches. Fr the cnstant e 3, all fur mdels predict nearly identical values fr the entire range f particle vlume fractins. Fr the piezelectric cnstant e 33, the Mri-Tanaka apprach exhibits the clsest agreement with the finite element predictins, especially fr particle vlume fractins abve 30%. The three dielectric cnstants fr the cmpsite are shwn in Fig. 10. Similar t the graphite/pdf cmpsite, all fur mdels predict identical values fr all three dielectric cnstants fr the entire range f particle vlume fractins. 3. PZT-7A/plyimide fiber cmpsite The Yung s mduli f the PZT-7A cmpsite are shwn in Fig. 11 fr the entire range f fiber vlume fractins. Fr the lngitudinal Yung s mdulus, Y 3, all three mdels predict the same values ver the cmplete range f vlume fractins. Fr the transverse Yung s mduli, Y 1 and Y, the Mri-Tanaka and finite element mdels have clse agreement up t a fiber vlume fractin f 80%. At a fiber vlume fractin f 90%, the prpsed mdel exhibits the clsest agreement t the finite element mdel. The Self-cnsistent methd significantly ver-predicts the finite element mdel at fiber vlume fractins abve 50%. The shear mduli f the cmpsite are pltted as a functin f fiber vlume fractin in Fig. 1. Fr the lngitudinal shear mduli, G 3 and G 13, and the transverse shear mdulus, G 1, the prpsed, Mri-Tanaka, and Self-cnsistent mdels ver-predict the finite element mdel fr the entire range f vlume fractins. While the Mri-Tanaka mdel exhibits the clsest agreement with the finite element mdel, the Self-cnsistent significantly ever-estimates the finite element mdel data. 10

11 The piezelectric cnstants f this material are shwn in Fig. 13. Fr the cnstants e 31 = e 3 and e 33, all fur mdels predict very similar values fr the entire range f fiber vlume fractins. Fr e 15, bth the prpsed and Mri-Tanaka mdels clsely agree with the finite element mdel fr the entire range f vlume fractins. The Self-cnsistent methd significantly ver-estimates e 15 fr the fiber vlume fractins abve 30%. The dielectric cnstants are shwn in Fig. 14 as a functin f fiber vlume fractin. Fr the transverse dielectric cnstants, κ 1 /κ 0 and κ /κ 0, the Mri-Tanaka mdel predicts the finite element mdel data better than des the prpsed and Self-cnsistent mdels fr fiber vlume fractins abve 50%. Fr the lngitudinal dielectric cnstant, κ 3 /κ 0, the fur mdels predict nearly identical values ver the entire range f vlume fractins. 4. PZT-7A/plyimide particle cmpsite The Yung s mduli f the PZT-7A/LaRC-SI particulate cmpsite are shwn in Fig. 15 as a functin f particle vlume fractin. Fr bth the transverse Yung s mduli, Y 1 and Y, and the lngitudinal Yung s mdulus, Y 3, the prpsed mdel agrees clsely with the finite element mdel fr the entire range f particle vlume fractins cnsidered. The Mri-Tanaka and Selfcnsistent mdels significantly under-predict and ver-predict, respectively, the finite element data fr vlume fractins abve 50%. The shear mduli f this material fr the range f vlume fractins are shwn in Fig. 16. Fr the lngitudinal shear mduli, G 3 and G 13, and the transverse shear mdulus, G 1, the prpsed, Mri-Tanaka, and Self-cnsistent mdels ver-predict the finite element mdel data fr the entire range f cnsidered particle vlume fractins, with the Mri-Tanaka exhibiting the clsest agreement. Fr particle vlume fractins abve 0%, the Self-cnsistent mdel significantly ver-predicts bth the lngitudinal and transverse shear mduli. The piezelectric cnstants are shwn in Fig. 17 as a functin f particle vlume fractin. Fr all fur cnstants, e 15, e 31 = e 3, and e 33, the finite element, prpsed, and Mri-Tanaka mdels shw clse agreement up t a particle vlume fractin f 40%. Fr the cnstants e 15 and e 3, the Mri-Tanaka mdel has the clsest agreement with the finite element mdel in the particle vlume fractin range between 40% and 50%. At a particle vlume fractin f 60%, the prpsed mdel exhibits the clsest agreement with the finite element mdel. Fr e 31 = e 3, the prpsed mdel shws the clsest match with the finite element mdel fr particle vlume fractins abve 40%. Fr particle vlume fractins abve 0%, the Self-cnsistent results are dramatic and d nt appear t clsely predict any f the piezelectric cnstants. The dielectric cnstants f the material are shwn in Fig. 18. Fr the transverse dielectric cnstants, κ 1 /κ 0 and κ /κ 0, and the lngitudinal dielectric cnstant, κ 3 /κ 0, the predicted values frm the Mri-Tanaka mdel agree with the finite element mdel up t a particle vlume fractin f abut 50%. Abve that value, the Mri-Tanaka mdel under-predicts the finite element data fr κ 1 /κ 0 and κ /κ 0. At that pint, the prpsed mdel exhibits better agreement with the finite element mdel. Fr particle vlume fractins abve 0%, the Self-cnsistent apprach significantly ver-estimates bth transverse and lngitudinal dielectric cnstants. 11

12 Summary and Cnclusins A new mdeling apprach has been prpsed fr predicting the bulk electrmechanical prperties f piezelectric cmpsites. The prpsed mdel ffers the same level f cnvenience as the Mri-Tanaka methd, that is, it des nt require iterative r numerical schemes fr btaining the predicted prperties, as is required with the Self-cnsistent and differential schemes. The electrmechanical prperties f fur piezelectric plymer cmpsite materials were predicted with the prpsed, Mri-Tanaka, Self-cnsistent methds, and detailed finite element analyses fr a wide range f matrix and reinfrcement electrmechanical prperties, gemetry, and vlume fractin. The fur piezelectric cmpsite materials cnsidered were: a graphite/pdf cmpsite, a SiC/PDF particulate cmpsite, a fibrus PZT- 7A/LaRC-SI cmpsite, and a PZT-7A/LaRC-SI particulate cmpsite. It was shwn that the prpsed mdel yields predicted prperties that were, in sme cases, mre accurate than the Mri-Tanaka and Self-cnsistent schemes. In particular, the prpsed mdel exhibits equal r clser agreement with the finite element mdel than des the Mri-Tanaka and Self-cnsistent schemes fr the predictin f several electrmechanical prperties. Fr the PDF matrix cmpsites these prperties include the lngitudinal shear and lngitudinal Yung s mduli and all dielectric cnstants fr the graphite/pdf cmpsite; and all Yung s mduli, all shear mduli (fr vlume fractins abve 50%), the piezelectric cnstants e 31 and e 3, and all dielectric cnstants fr the SiC/PDF cmpsite. Fr the PZT-reinfrced cmpsites these include the lngitudinal Yung s mdulus, the piezelectric cnstants e 33 and e 15, and the lngitudinal dielectric cnstant f the fibrus PZT-7A/LaRC-SI cmpsite; and all f the Yung s mduli, all f the piezelectric cnstants, and the lngitudinal dielectric cnstant (fr vlume fractins abve 60%) f the particulate PZT-7A/LaRC-SI cmpsite. Based n these results, the chice f the mst accurate mdel (between the prpsed, Mri-Tanaka, and Selfcnsistent methds) fr a specific piezelectric cmpsite material shuld be based n the cnstituent prperties and the gemetry and vlume fractin f the inclusins. 1

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15 [37] Christensen, R.M.: A Critical Evaluatin fr a Class f Micrmechanics Mdels. Jurnal f the Mechanics and Physics f Slids. l. 38, 1990, pp [38] Christensen, R.M.; Schantz, H., and Shapir, J.: On the Range f alidity f the Mri- Tanaka Methd. Jurnal f the Mechanics and Physics f Slids. l. 40, 199, pp [39] Benedikt, B.; Rupnwski, P., and Kumsa, M.: isc-elastic Stress Distributins and Elastic Prperties in nidirectinal Cmpsites with Large lume Fractins f Fibers. Acta Materialia. l. 51, N. 1, 003, pp [40] Halpin, J.C. and Tsai, S.W.: Envirnmental Factrs in Cmpsites Design. AFML-TR , [41] Davis, P.J. and Plnsky, I.: Numerical Interplatin, Differentiatin and Integratin. In: Handbk f Mathematical Functins with Frmulas, Graphs, and Mathematical Tables. M. Abramwitz and I. A. Stegun, eds. New Yrk: Jhn Wiley & Sns, 197, pp [4] Sun, C.T. and aidya, R.S.: Predictin f Cmpsite Prperties frm a Representative lume Element. Cmpsites Science and Technlgy. l. 56, 1996, pp

16 Table 1. Electrmechanical prperties f matrix and inclusin materials Prperty PDF LaRC-SI Graphite SiC PZT-7A fiber particle C 11 (GPa) C 1 (GPa) C 13 (GPa) C (GPa) C 3 (GPa) C 33 (GPa) C 44 (GPa) C 55 (GPa) C 66 (GPa) κ 1 /κ κ /κ κ 3 /κ e 15 (C/m ) e 31 (C/m ) e 3 (C/m ) e 33 (C/m )

17 Table. Bundary cnditins fr axial stiffness cmpnents Prperty Applied strain and electric field Displacements and electric ptential Elastic energy u B = ε x C ε = ε C ε =ε C ε = ε u u φ u u B = ε x u φ u u u B = ε x 3 3 φ e e e = C11 ε = C ε = C33 ε Table 3. Bundary cnditins fr plane-strain bulk mduli Prperty Applied strain and electric field Displacements and electric ptential Elastic energy u B K 3 33 K K ε =ε =ε u ε =ε = ε B ε =ε =ε u u B = ε x 3 = ε 3 u B x φ u B = ε x 1 1 u B = ε x 3 3 φ u B = ε x 1 1 = ε 3 u B x φ e e e = K3 ε = K13 ε = K1 ε 17

18 Table 4. Bundary cnditins fr shear stiffness cmpnents Prperty Applied strain and electric field Displacements and electric ptential Elastic energy u B C 44 C 55 C 66 γ ε 3 = γ ε 13 = γ ε 1 = 1 ( ) = ( γ ) ( ) = ( γ ) u B x 3 u B x 3 φ = ( γ ) u B x 1 3 u = ( γ ) 3 1 u B x φ ( ) = ( γ ) ( ) = ( γ ) u B x 1 u B x 1 u 3 φ e e e = C44 γ = C55 γ = C66 γ Table 5. Bundary cnditins fr dielectric cnstants Applied strain Displacements Dielectric Prperty and and energy electric field electric ptential u1 u d = κ1 E κ 1 /κ 0 E1 = E u B κ /κ 0 E = E κ 3 /κ 0 E = E 3 3 u1 u u3 u1 u u3 φ = φ φ = = Ex1 Ex Ex3 d d = κ = κ 3 ( E ) ( E ) 18

19 Table 6. Bundary cnditins fr piezelectric cnstants Applied strain Displacements Electrmechanical Prperty and and energy electric field electric ptential u1 = ( γ ) x3 γ ε 13 = u e 15 em = e15γ E E1 = E u3 = ( γ ) x 1 φ B = Ex ε =ε 11 e 31 E3 = E ε =ε e 3 E3 = E ε =ε 33 e 33 E3 = E u u3 u1 u3 u1 u u B =ε x φ 1 1 = 1 Ex3 u B =ε x φ = Ex3 u B =ε x φ 3 3 = Ex3 em = e31ε E em = e3ε E em = e33ε E 19

20 Plymer bundary Fiber RE Full finite element mdel Figure 1. Finite element RE f fiber cmpsite 0

21 Plymer Particle bundary RE Full finite element mdel Figure. Finite element RE f particle cmpsite 1

22 30 Yung's mdulus (GPa) Finite element, Y 1 Finite element, Y Finite element, Y 3 Prpsed, Y 1 Prpsed, Y Prpsed, Y 3 Mri-Tanaka, Y 1 Mri-Tanaka, Y Mri-Tanaka, Y 3 Self-Cnsistent, Y 1 Self-Cnsistent, Y Self-Cnsistent, Y Fiber vlume fractin (%) Figure 3. Yung s mduli vs. fiber vlume fractin fr graphite/pdf cmpsite 5 Shear mdulus (GPa) Finite element, G 3 Finite element, G 13 G 1 Prpsed, G 3 Prpsed, G 13 G 1 Mri-Tanaka, G 3 Mri-Tanaka, G 13 G 1 Self-Cnsistent, G 3 Self-Cnsistent, G 13 G Fiber vlume fractin (%) Figure 4. Shear mduli vs. fiber vlume fractin fr graphite/pdf cmpsite

23 Piezelectric cnstant (C/m ) Finite element, e 31 Finite element, e 3 Finite element, e 33 Prpsed, e 31 Mri-Tanaka, e 31 Mri-Tanaka, e 3 Mri-Tanaka, e 33 Self-Cnsistent, e Prpsed, e 3 Prpsed, e 33 Self-Cnsistent, e 3 Self-Cnsistent, e Fiber vlume fractin (%) Figure 5. Piezelectric cnstants vs. fiber vlume fractin fr graphite/pdf cmpsite 1 11 Dielectric cnstant Finite element, κ 1 /κ 0 Mri-Tanaka, κ 1 /κ 0 Finite element, κ /κ 0 Mri-Tanaka, κ /κ 0 Finite element, κ 3 /κ 0 Mri-Tanaka, κ 3 /κ 0 Prpsed, κ 1 /κ 0 Self-Cnsistent, κ 1 /κ 0 Prpsed, κ /κ 0 Self-Cnsistent, κ /κ 0 Prpsed, κ 3 /κ 0 Self-Cnsistent, κ 3 /κ Fiber vlume fractin (%) Figure 6. Dielectric cnstants vs. fiber vlume fractin fr graphite/pdf cmpsite 3

24 35 Yung's mdulus (GPa) Finite element, Y 1 Finite element, Y Finite element, Y 3 Prpsed, Y 1 Prpsed, Y Prpsed, Y 3 Mri-Tanaka, Y 1 Mri-Tanaka, Y Mri-Tanaka, Y 3 Self-Cnsistent, Y 1 Self-Cnsistent, Y Self-Cnsistent, Y Particle vlume fractin (%) Figure 7. Yung s mduli vs. particle vlume fractin fr SiC/PDF cmpsite 0 Shear mdulus (GPa) Finite element, G 3 Finite element, G 13 Finite element, G 1 Prpsed, G 3 Prpsed, G 13 Prpsed, G 1 Mri-Tanaka, G 3 Mri-Tanaka, G 13 Mri-Tanaka, G 1 Self-Cnsistent, G 3 Self-Cnsistent, G 13 Self-Cnsistent, G Particle vlume fractin (%) Figure 8. Shear mduli vs. particle vlume fractin fr SiC/PDF cmpsite 4

25 Piezelectric cnstant (C/m ) Finite element, e 31 Finite element, e 3 Finite element, e 33 Prpsed, e 31 Prpsed, e 3 Prpsed, e 33 Mri-Tanaka, e 31 Mri-Tanaka, e 3 Mri-Tanaka, e 33 Self-Cnsistent, e 31 Self-Cnsistent, e 3 Self-Cnsistent, e Particle vlume fractin (%) Figure 9. Piezelectric cnstants vs. particle vlume fractin fr SiC/PDF cmpsite 10 9 Dielectric cnstant 8 7 Finite element, κ 1 /κ 0 Finite element, κ /κ 0 Finite element, κ 3 /κ 0 Mri-Tanaka, κ 1 /κ 0 Mri-Tanaka, κ /κ 0 Mri-Tanaka, κ 3 /κ 0 Prpsed, κ 1 /κ 0 Self-Cnsistent, κ 1 /κ 0 Prpsed, κ /κ 0 Self-Cnsistent, κ /κ 0 Prpsed, κ 3 /κ 0 Self-Cnsistent, κ 3 /κ Particle vlume fractin (%) Figure 10. Dielectric cnstants vs. particle vlume fractin fr SiC/PDF cmpsite 5

26 80 Yung's mdulus (GPa) Finite element, Y 1 = Y Finite element, Y 3 Prpsed, Y 1 = Y Prpsed, Y 3 Mri-Tanaka, Y 1 = Y Mri-Tanaka, Y 3 Self-Cnsistent, Y 1 = Y Self-Cnsistent, Y Fiber vlume fractin (%) Figure 11. Yung s mduli vs. fiber vlume fractin fr PZT-7A/LaRC-SI cmpsite 30 Shear mdulus (GPa) Finite element, G 3 = G 13 Finite element, G 1 Prpsed, G 3 = G 13 Prpsed, G 1 Mri-Tanaka, G 3 = G 13 Mri-Tanaka, G 1 Self-Cnsistent, G 3 = G 13 Self-Cnsistent, G Fiber vlume fractin (%) Figure 1. Shear mduli vs. fiber vlume fractin fr PZT-7A/LaRC-SI cmpsite 6

27 Piezelectric cnstant (C/m ) Finite element, e 15 Finite element, e 31 = e 3 Finite element, e 33 Prpsed, e 15 Prpsed, e 31 = e 3 Prpsed, e 33 Mri-Tanaka, e 15 Mri-Tanaka, e 31 = e 3 Mri-Tanaka, e 33 Self-Cnsistent, e 15 Self-Cnsistent, e 31 = e 3 Self-Cnsistent, e Fiber vlume fractin (%) Figure 13. Piezelectric cnstants vs. fiber vlume fractin fr PZT-7A/LaRC-SI cmpsite 50 Dielectric cnstant Finite element, κ 1 /κ 0 = κ /κ 0 Finite element, κ 3 /κ 0 Prpsed, κ 1 /κ 0 = κ /κ 0 Prpsed, κ 3 /κ 0 Mri-Tanaka, κ 1 /κ 0 = κ /κ 0 Mri-Tanaka, κ 3 /κ 0 Self-Cnsistent, κ 1 /κ 0 = κ /κ 0 Self-Cnsistent, κ 3 /κ Fiber vlume fractin (%) Figure 14. Dielectric cnstants vs. fiber vlume fractin fr PZT-7A/LaRC-SI cmpsite 7

28 60 Yung's mdulus (GPa) Finite element, Y 1 = Y Finite element, Y 3 Prpsed, Y 1 = Y Prpsed, Y 3 Mri-Tanaka, Y 1 = Y Mri-Tanaka, Y 3 Self-Cnsistent, Y 1 = Y Self-Cnsistent, Y Particle vlume fractin (%) Figure 15. Yung s mduli vs. particle vlume fractin fr PZT-7A/LaRC-SI cmpsite 5 Shear mdulus (GPa) Finite element, G 3 = G 13 Finite element, G 1 Prpsed, G 3 = G 13 Prpsed, G 1 Mri-Tanaka, G 3 = G 13 Mri-Tanaka, G 1 Self-Cnsistent, G 3 = G 13 Self-Cnsistent, G Particle vlume fractin (%) Figure 16. Shear mduli vs. particle vlume fractin fr PZT-7A/LaRC-SI cmpsite 8

29 0.90 Piezelectric cnstant (C/m ) Finite element, e 15 Finite element, e 31 = e 3 Finite element, e 33 Prpsed, e 15 Prpsed, e 31 = e 3 Prpsed, e 33 Mri-Tanaka, e 15 Mri-Tanaka, e 31 = e 3 Mri-Tanaka, e 33 Self-Cnsistent, e 15 Self-Cnsistent, e 31 = e 3 Self-Cnsistent, e Particle vlume fractin (%) Figure 17. Piezelectric cnstants vs. particle vlume fractin fr PZT-7A/LaRC-SI cmpsite Dielectric cnstant Finite element, κ 1 /κ 0 = κ /κ 0 Finite element, κ 3 /κ 0 Prpsed, κ 1 /κ 0 = κ /κ 0 Prpsed, κ 3 /κ 0 Mri-Tanaka, κ 1 /κ 0 = κ /κ 0 Mri-Tanaka, κ 3 /κ 0 Self-Cnsistent, κ 1 /κ 0 = κ /κ 0 Self-Cnsistent, κ 3 /κ Particle vlume fractin (%) Figure 18. Dielectric cnstants vs. particle vlume fractin fr PZT-7A/LaRC-SI cmpsite 9