The Effective Pyroelectric and Thermal Expansion Coefficients Of

Mechanics of Materials 36 (2004) 949–958 www.elsevier.com/locate/mechmat

The eﬀective pyroelectric and thermal expansion coeﬃcients of ferroelectric ceramics JiangYu Li * Department of Engineering Mechanics, University of Nebraska-Lincoln, W317.5 Nebraska Hall, Lincoln, NE 68588-0526, USA Received 5 November 2002; received in revised form 13 May 2003

Abstract

We present a micromechanical analysis on the effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics in terms of their microstructural information. The overall behaviors of ferroelectric ceramics are profoundly influenced by the microstructural phenomena, where the macroscopic pyroelectric effect can be induced in an originally isotropic, thus non-pyroelectric ceramic composed of randomly oriented ferroelectric grains through poling, during which the polar axes of grains are switched by the applied electric field or mechanical stress. To analyze these complicated phenomena, we will first establish an exact connection between the effective thermal moduli and the effective electroelastic moduli of ferroelectric ceramics, and then combine the exact connection with the effective medium approximation to provide an estimate on the effective pyroelectric and thermal expansion coefficients of ferroelectric ceramics in terms of the orientation distribution of grains and poling conditions, where the texture evolution as a result of domain switching during poling has been taken into account. Numerical results are presented and good agreements with known theoretical results and some experimental data are observed. Ó 2003 Elsevier Ltd. All rights reserved.

Keywords: Pyroelectricity; Ferroelectricity; Piezoelectricity; Ceramics; Eﬀective moduli

1. Introduction recent years, and have been utilized for various application such as infrared detection, imaging The crystals having a spontaneous polarization systems, and thermal-medical diagnostics (Cross, along a unique polar axis are called pyroelectric 1993). (Jona and Shirane, 1993), where the spontaneous A ceramic made of pyroelectric grains does not polarization is dependent on the temperature, and necessarily possess overall pyroelectric effect, be- electric charges can be induced on the crystal cause the center-symmetry of randomly oriented surface by temperature change. Pyroelectric de- ceramics prevents pyroelectric effect at macro- vices relying on the temperature sensitivity of scopic scale. If the grains are also ferroelectric, i.e., polarization have received increasing interest in if the polar axis can be switched by the applied electric field or mechanical stress, then macroscopic pyroelectric effect can be induced in the * Tel.: +1-402-4721631; fax: +1-402-4728292. ceramics through poling, where a high electric E-mail address: [email protected] (J.Y. Li). field is applied to the ceramic at an elevated

0167-6636/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2003.05.005 950 J.Y. Li / Mechanics of Materials 36 (2004) 949–958 temperature. The microstructural evolution during governing the effective constitutive moduli in Sec- poling is very complex and involves polarization tion 3. Micromechanics approximation will be switching, domain wall movement, intergranular introduced in Section 4, and microstructural fea- constraint, and microstress generation. The com- tures such as texture and orientation distribution plicated microstructural phenomena obviously function (ODF) under various poling conditions have profound influence on the macroscopic are discussed in Section 5. We will then present pyroelectricity of ceramics, and this work will some numerical results and discussion on the study the effects of microstructural phenomena at effective pyroelectric and thermal expansion coef- domain and grain levels, including orientation ficients of ferroelectric ceramics in Section 6. It is distribution of domains and grains, on the effective demonstrated that the microstructure has impor- pyroelectric and thermal expansion coefficients of tant effect on the overall pyroelectric and thermal ferroelectric ceramics. The texture evolution in expansion coefficients of ceramics, and good ferroelectric ceramics during poling will be taken agreements between numerical results and known into account. theoretical results and some experimental data are Although much effort has been made to under- observed. stand the effective piezoelectric moduli of ferroelectric ceramics (Dunn, 1995; Nan and Clarke, 1996; Rodel€ and Kreher, 1999; Li, 2000a), little work has 2. Basic equations and notation been done to predict the effective thermal moduli of ferroelectric ceramics in terms of their microstruc- We consider the piezoelectric analog of the ture. Instead, quite a few micromechanical analysis uncoupled theory of thermoelasticity, where the on the effective thermal moduli of piezoelectric constitutive equations are given by composites have been developed (Dunn, 1993a,b; E E Benveniste, 1994; Chen, 1994a; Qin et al., 1998; Li rij ¼ L ekl þ eijkðÀEkÞÀk h; ijkl ij ð1Þ and Dunn, 1998; Levin et al., 1999; Aboudi, 2001). e e Di ¼ eiklekl À j ðÀEkÞÀp h This is probably because the microstructural phe- ik i nomena in ferroelectric ceramics are more in- or its inverse volved. In this work, we extend the self-consistent D D approach originally developed for electroelastic eij ¼ Mijklrkl þ gijkDk þ aij h; ð2Þ moduli (Li, 2000a) to predict the effective pyro- r r À Ei ¼ giklrkl À tikDk þ ci h: electric and thermal expansion coefficients of ferroelectric ceramics. The analysis shares the same In the equations, rij and ekl are stress and strain spirit as many works in elastic polycrystals, where tensors, respectively; Di and Ek are electric dis- an exact connection between the effective thermal placement and electric field, respectively; h is the moduli and the effective electroelastic moduli is temperature change with respect to a reference E established first (Levin, 1967; Rosen and Hashin, temperature. The meanings of elastic moduli Lijkl D 1970; Hashin, 1984; Schulgasser, 1987; Kreher, and Mijkl, piezoelectric coefficients eijk and gijk, e r 1988; Li, 2000b), and the effective medium ap- dielectric constants jik and tik, thermoelastic con- E D e proximation is then applied to analyze the local stants kij and aij , and thermoelectric constants pi r electromechanical fields in ceramics (Walpole, and ci are clear from the constitutive equations. 1969; Willis, 1977; Qiu and Weng, 1991). The re- We notice that ÀEi rather than Ei is used as cent results on the equilibrium domain configura- dependent variable, because it gives us the sym- tion in a saturated ferroelectric ceramics have been metric moduli matrix which proves to be conve- employed to estimate the effective texture of poled nient. We also notice that by choosing different ferroelectric ceramics (Bhattacharya and Li, 2001; sets of independent variables, other sets of con- Li and Bhattacharya, 2002). stitutive equations can be obtained. For example, We introduce the basic equation and notation it is easier to measure thermal expansion coeffi- o in Section 2, followed by the exact relationships cients aE eij and pyroelectric coefficients ij ¼ oh jrmn;Em J.Y. Li / Mechanics of Materials 36 (2004) 949–958 951

r oDi p ¼ j , and they are related to current À1 i oh rmn;En H ¼ G ; C ¼ H K : ð8Þ moduli by JilK lKJi Ji JilK lK Finally, we notice the differential constraints on aD ¼ aE À g pr; cr ¼ tr pr: ð3Þ ij ij ijk k i ij j the field variables in terms of the equilibrium r Notice that ci is the figure of merit used by a de- equation signer to assess the performance of a pyroelectric RiJ;i ¼ 0; ð9Þ material for a typical device (Newnham et al., 1978). and gradient equation

To proceed, we adopt the notation introduced ZJi ¼ UJ;i; ð10Þ by Barnett and Lothe (1975) that treats the elastic and electric variables on an equal footing. It is with similar to the conventional indicial notation with uj; J ¼ 1; 2; 3; the exception that both lowercase and uppercase UJ ¼ ð11Þ /; J ¼ 4; subscripts are used as indices, where lowercase ones take on the range 1–3, and uppercase ones where ui and / are displacement and electric po- take on the range 1–4, and repeated uppercase tential, respectively. subscripts are summed over 1–4. With this notation, the field variables are expressed as 3. The effective moduli rij; J ¼ 1; 2; 3; RiJ ¼ Di; J ¼ 4; ð4Þ Let us now consider a ferroelectric polycrys- ekl; K ¼ 1; 2; 3; talline ceramic subjected to a uniform temperature ZKl ¼ ÀEl; K ¼ 4; change h, with linear displacement and electric 0 potential ZKlxl, or uniform traction and electric the electroelastic moduli are expressed as 0 8 displacement RkLnk applied at the boundary, where > LE ; J; K ¼ 1; 2; 3; x is the position vector and n is the outward > ijkl l k < normal. The polycrystal is composed of grains eijl; J ¼ 1; 2; 3; K ¼ 4; GiJKl ¼ with certain orientation distribution, each having > e ; J ¼ 4; K ¼ 1; 2; 3; :> ikl distinct constitutive moduli in the global coordi- Àje ; J; K ¼ 4; nate system due to the variation in grainsÕ orien- 8 il ð5Þ D tation and grain anisotropy. The constitutive > Mijkl; J; K ¼ 1; 2; 3; <> moduli of polycrystalline aggregate, as a result, is gijl; J ¼ 1; 2; 3; K ¼ 4; HJilK ¼ heterogenous at microscopic level. This leads to a > g ; J 4; K 1; 2; 3; :> ikl ¼ ¼ heterogeneous local electromechanical field in r Àtil; J; K ¼ 4 polycrystal at grain level, which depends on X, the orientation of individual grains. However, at the and the thermal moduli are expressed as ( macroscopic level the effective constitutive moduli E kij; J ¼ 1; 2; 3; can be defined through the effective constitutive KiJ ¼ e equation if the polycrystal is statistically homo- pi ; J ¼ 4; ( ð6Þ geneous (Nemat-Nasser and Hori, 1999), aD; J ¼ 1; 2; 3; C ij Ã Ã Ji ¼ r hRiJ ðXÞi ¼ GiJKlhZKlðXÞi À KiJ h; ci ; J ¼ 4: ð12Þ Ã Ã hZKlðXÞi ¼ HKliJ hRiJ ðXÞi þ CKlh; As a result, the constitutive equations (1) and (2) can be rewritten as where hÁi is used to denote the orientation-aver- aged quantities in polycrystal, weighted by the R ¼ G Z À K h; Z ¼ H R þ C h; ð7Þ iJ iJKl Kl iJ Ji JilK lK Ji orientation distribution function (ODF), which where describes the probability of a grain falling into 952 J.Y. Li / Mechanics of Materials 36 (2004) 949–958 certain range of orientation. From the averaging and thermal concentration factors aJiðXÞ and theorem (Dunn and Taya, 1993), we have biJ ðXÞ defined by 0 0 II II hZKlðXÞi ¼ ZKl or hRiJ ðXÞi ¼ RiJ ; ð13Þ ZJi ðXÞ¼aJiðXÞh; RiJ ðXÞ¼biJ ðXÞh; ð19Þ for specified linear displacement and electric po- which leads to tential or uniform traction and electric displace- GÃ ¼hG ðXÞA ðXÞi; ment at the boundary. iJKl iJNm NmKl Ã ð20Þ In order to derive the effective constitutive HJilK ¼hHJimN ðXÞBmNlK ðXÞi moduli of ferroelectric polycrystals, it is conve- and nient to decompose the electroelastic field in the Ã polycrystal into two parts, one is due to the elec- KiJ ¼hÀGiJNmðXÞaNmðXÞþKiJ ðXÞi; Ã ð21Þ tromechanical loading (the applied linear dis- CJi ¼hHJimN ðXÞbmN ðXÞþCJiðXÞi: placement and potential Z0 x or the applied Ji i The determination of the effective constitutive uniform traction and electric displacement R0 n at iJ i moduli of polycrystal is thus dependent on the the boundary), the loading field I, and the other is determination of field and thermal concentration due to the temperature change h, the thermal field factors as functions of grainsÕ orientation, and II. This decomposition is possible because of the their orientational averaging. We also notice from linearity of the constitutive relationships. In light the averaging theorem that of this decomposition, the effective constitutive equation for loading field and thermal field can be hAiJlK ðXÞi ¼ IiJlK ; hBJiKlðXÞi ¼ IJiKl ð22Þ written as and I Ã 0 I Ã 0 hRiJ ðXÞi ¼ GiJKlZKl or hZJiðXÞi ¼ HJilK RlK haJiðXÞi ¼ 0; hbiJ ðXÞi ¼ 0; ð23Þ ð14Þ with 8 and > i ; J; K ¼ 1; 2; 3; <> ijkl 0; J ¼ 1; 2; 3; K ¼ 4; hRII ðXÞi ¼ ÀKÃ h; hZIIðXÞi ¼ CÃ h; ð15Þ iJ iJ Ij Ij IiJlK ¼ IJiKl ¼ > ð24Þ :> 0; J ¼ 4; K ¼ 1; 2; 3; whereas the constitutive equations for individual dil; J; K ¼ 4; grains can be written as where iijkl is the forth-order unit tensor, and dkm is I I RiJ ðXÞ¼GiJKlðXÞZKlðXÞ; the Kronecker delta. ð16Þ It is also possible to determine the effective ZI ðXÞ¼H ðXÞRI ðXÞ Ji JilK lK thermal moduli of polycrystals using field con- and centration factors instead of thermal concentra- II II tion factors. To this end, we recall the generalized RiJ ðXÞ¼GiJKlðXÞZKlðXÞÀCiJ ðXÞh; ð17Þ Hill condition for piezoelectric solids (Li and II II ZJi ðXÞ¼HJilK ðXÞRlK ðXÞþKJiðXÞh: Dunn, 1999),

It is noted that at the microscopic level, the ther- hRiJ ðXÞZJiðXÞi ¼ hRiJ ðXÞihZJiðXÞi; ð25Þ II II mal field RiJ ðXÞ or ZJi ðXÞ has contribution from which is valid for uniform traction and electric both electroelastic field and temperature change, displacement or linear displacement and electrical due to the interaction between grains in polycrys- potential boundary condition. The electrome- tals. Because of the linearity, we can introduce the chanical fields RiJ and ZJi do not need to be con- field concentration factors AJiKlðXÞ and BkLmN ðXÞ nected by certain constitutive equations. Now defined by I consider the loading field ZJi due to the linear I 0 elastic displacement and potential boundary con- ZJiðXÞ¼AJiKlðXÞZKl or II dition, and thermal field RiJ due to the tempera- I 0 RiJ ðXÞ¼BiJlK ðXÞRlK ; ð18Þ ture change, we have J.Y. Li / Mechanics of Materials 36 (2004) 949–958 953

II I II 0 Ã 0 yet to be determined uniform eﬀective moduli to RiJ ðXÞZJiðXÞ ¼ RiJ ðXÞ ZJi ¼ÀKiJ hZJi ð26Þ simulate the electromechanical ﬁeld in a grain at a due to the generalized Hill condition. The left particular orientation in a ceramic. This approxi- hand side of Eq. (26) can be written as mation leads to

II I II dil dil À1 R ðXÞZ ðXÞ ¼ðGiJKlðXÞZ ðXÞ A X A X A X 30 iJ Ji Kl JiKlð Þ¼ JiNmð Þh KlNmð Þi ð Þ À K ðXÞhÞZI ðXÞ iJ Ji with ¼ RI ðXÞZII ðXÞ lK Kl Adil ðXÞ¼½I þS ðXÞH Ã ðG ðXÞÀGÃ ÞÀ1; I JiKl KlJi KlMn MnoP oPJi oPJi À KiJ ðXÞZ ðXÞh Ji ð31Þ ¼À K ðXÞZI ðXÞh iJ Ji 0 derived from the single inclusion solution, where ¼À KiJ ðXÞAJiKlðXÞZ h : ð27Þ Kl SKlMn are the piezoelectric Eshelby tensors which In the derivation, we have used the symmetry of are functions of grain shape, orientation, and the II electromechanical moduli of matrix (Dunn and GiJKl and the fact that hZJi i¼0 under the specified boundary condition. Combining Eqs. (26) and Taya, 1993). In general, Eq. (30) need to be solved (27), we obtain numerically by iteration, which usually converges pretty fast. Notice that the interaction between KÃ ¼hA ðXÞK ðXÞi: ð28Þ iJ KlJi lK grains is taken into account in the self-consistent A similar manipulation for the applied traction approach through the effective medium approxi- and electric displacement boundary condition mation, and the normalization condition (22) is yields automatically satisfied. A simpler approximation Ã would assume CJi ¼hBlKiJ ðXÞCKlðXÞi: ð29Þ Thus the effective thermal moduli can be deter- AJiKl ¼ IJiKl; BiJlK ¼ IiJlK ; ð32Þ mined from either thermal concentration factors which is analog of Voigt and Reuss averages for or field concentration factors. Notice that in gen- elastic materials. The Voigt and Reuss aver- A B eral the concentration factors JiKl and iJlK do not ages usually do not predict the effective moduli have diagonal symmetry. of polycrystal accurately when the crystalline anisotropy is strong.

4. The micromechanics approximations 5. Orientation distribution function and texture Up to Eq. (29) the derivation is rigorous under the assumption of macroscopic homogeneity. To The issue remains on how to characterize the carry out the analysis further, we need to make microstructure of ferroelectric ceramics. The ori- approximations regarding the microstructural field entation of grains in a ferroelectric ceramic could distribution in the polycrystal, and micromechan- be described by hg, the angle between the ics approach is appropriate. In this paper we rely global poling axis of ceramic and the local crys- the determination of the effective thermal moduli tallographical axis of grains. Within each grains on the field concentration factors, i.e., Eqs. (28) there are numerous domains whose orientation and (29). The key step in the prediction of the can be described by hdðhgÞ, the angle between the effective moduli thus is the determination of the polarization direction of domains and the poling field concentration factor AJiKl and BiJlK . To this direction of ceramic, which depends on the grainÕs end we adopt a self-consistent approach suitable orientation hg through crystallographic con- for piezoelectric polycrystals. The essence of self- straints. The number of possible domain orienta- consistent approach is using the electromechanical tions within a particular grain is governed field in a single grain embedded in a matrix with by crystallography. For example, in tetragonal 954 J.Y. Li / Mechanics of Materials 36 (2004) 949–958 system, six orientations are available while for and only part of 180° domains are switched. As a rhombohedral system, there are eight possible result, the Ôfully poledÕ ceramics have most domain domain orientations in each grain. The statistical switched, and Taylor estimated ceramics have least distribution of hg and hd are then described by the domain switched. The change in orientation dis- g orientation distribution function (ODF) W ðhgÞ tribution of domains leads to macroscopic pyro- d and W ðhdÞ, which gives the probability of finding electricity in ceramics, which we analyze here. a grain or a domain in a particular orientation. In In addition to the texture evolution due to the the as-processed states, individual grains are ran- domain switching induced by poling, the pre-pol- g domly oriented in ceramics, so that W ðhgÞ and ing texture of ceramics also have important influ- d W ðhdÞ are uniform throughout the ceramics and ences on the overall behaviors of ceramics. To independent on hg and hd, and the ceramic is iso- analyze this effect we adopt Gaussian distribution tropic and non-pyroelectric. The poling process, function as ODF of grains for pre-poling texture, where a high electric field is applied to ceramics at ! h2 elevated temperature, tends to realign the polari- 1 g W ðhgÞ¼ pffiffiffiffiffiffi exp À ; ð34Þ zation directions of domains as closely as possible l 2p 2l2 to the applied field, thus effectively changes the d which can be used to simulate a wide range of orientation distribution of domains, W ðhdÞ, but does not change the orientation distribution of texture in piezoelectric polycrystals by adjusting g the parameter l. For example l !1 represents grains, W ðhgÞ. For ceramics composed of randomly oriented tetragonal grains, if all domains random texture in unpoled isotropic ceramics, and are switched to be as closely as possible to the l ! 0 represents fiber texture in thin films. The d ODF for various textures in ferroelectric ceramics poling direction in a Ôfully poledÕ state, W ðhdÞ are summarized in Table 1. They need to be nor- would be uniform for hd 2½0; p=4 and zero elsewhere. If all 180° domains are switched and none malized to be used in the orientational averaging of the 90° domains are switched in a Ôpartially of any physical quantities of polycrystals. poledÕ state, the corresponding ODF for domains is uniform for hd 2½0; p=2 and zero elsewhere. The actual orientation distribution of domains in a 6. Numerical results and discussion poled ferroelectric ceramic is rather difficult to determine, but can be fairly well approximated by With ODF given, we developed a numerical Taylor texture derived from Taylor estimate (Li algorithm to carry out the orientational averaging and Bhattacharya, 2002), in Eqs. (28) and (29) by numerical integration using Gaussian quadrature method (Press et al., 1 þ cos hd W ðhdÞ¼ : ð33Þ 1992), where the integral is approximated by the 2 sum of its integrand values at a set of points called This expression is derived based on the assumption abscissas weighted by weighting coefficients. We that strain and polarization caused by the domain carried out the numerical calculations for tetrag- switching is uniform in polycrystal, which leads to onal barium titanate ceramics, with materials data accurate predictions on the saturation strain and of single domain single crystal listed in Table 2. polarization in ferroelectric ceramics. In the Tay- The electroelastic moduli are obtained from lor estimate, none of 90° domains are switched, Zgonik et al. (1994), and the thermal moduli are

Table 1 Orientation distribution function of polycrystals with different textures and poling conditions Texture Random Random Gaussian Taylor Poling Full Partial Partial Partial 2 W h 1 h 0; p=4 1 h 0; p=2 p1ffiffiffiffi exp h 1þcos h ð Þ ð 2½ Þ ð 2½ Þ l 2p À 2l2 2 J.Y. Li / Mechanics of Materials 36 (2004) 949–958 955

Table 2 The thermoelectroelastic moduli of barium titanate single crystal (Zgonik et al., 1994; Jona and Shirane, 1993) E E E E E E e e E E r L11 L12 L13 L33 L44 L66 e31 e33 e15 j11 j33 a11 a33 p3 222 108 111 151 61 134 )0.7 6.7 34.2 2200 56 11.8 )18.5 5.53 2 À6 À4 2 L: GPA, e: C/m , j: j0, a: 10 , p: 10 C/m K.

D estimated from figures in (Jona and Shirane, 1993). domain switching tends to decrease a33 and in- D In the calculation, the grain is assumed to be crease a11, consistent with the anisotropy of ther- spherical. mal expansion coefficients of single crystals. Poor We first demonstrate the effect of poling on the agreement with experiment measurement on ther- effective thermal moduli. The effective pyroelectric mal expansion coefficients are observed. This is at and thermal expansion coefficients are given in least partly due to the lacking of accurate single Figs. 1 and 2in terms of various degree of poling, crystal data. and are compared with experiment measurement. We then demonstrate the effect of textures on It is clear that domain switching tends to increase the effective pyroelectric and thermal expansion the effective pyroelectric coefficient, and Taylor coefficients using Gaussian distribution function as estimate on pyroelectric coefficient agrees with ODF. The effective pyroelectric and thermal experiment measurement well, which suggests that expansion coefficients in terms of Gaussian distri- only some 180° domains and few of 90° domains bution parameter l predicted by the self-consistent are switched during poling. This is consistent with approach and Voigt–Reuss averages are given in well known observation that 90° domain switching Figs. 3 and 4. We also give the effective piezo- is much harder than 180° domain switching, which electric coefficients in Fig. 5 as a comparison. does not involve transformation strain as 90° do- When l approaches zero, which implies that the main switching does. It is also observed that Ôfully ceramics posses fiber texture, the predictions of D poledÕ ceramic has highest a11, while Ôpartially pyroelectric coefficients, piezoelectric coefficients, D poledÕ ceramic has highest a33. It suggests that 90° and thermal expansion coefficients given by the

Fig. 1. The effective pyroelectric coefficient for various extents Fig. 2. The effective thermal expansion coefficients for various of poling; experiment data are from Dunn (1993a,b). extents of poling; experiment data are from Dunn (1993a,b). 956 J.Y. Li / Mechanics of Materials 36 (2004) 949–958

5 ×105 40 The self-consistent approach e The Self-Consistent Approach 15 Voigt-Reuss averages Voigt-Reuss Averages

5 Experiment (N/CK) 4 ×10 30 Experiment ) σ 2 3 γ (C/m ij e 3 ×105 20 33

2 ×105 10 e 33 e 15

5 Piezoelectric Constants e 1 ×10 0 e e 31

The effective pyroelectric coefficient - 31

0 -10 0.01 0.1 1 10 100 0.01 0.1 1 10 100 µ µ

Fig. 3. The effective pyroelectric coefficient for various textures; Fig. 5. The effective piezoelectric coefficients for various tex- experiment data are from Dunn (1993a,b). tures; experiment data are from Dunn (1993a,b).

2.5 ×10-5 pyroelectric and piezoelectric coefficients become The self-consistent approach α zero, and a33 converges to a11, as expected. The (1/K) 33 Voigt-Reuss averages D ij × -5 effective pyroelectric coefficient decreases with the α 2 10 Experiment increase of l monotonically, while the variations of thermal expansion coefficients and piezoelectric × -5 1.5 10 constants are not monotonic. It is also observed that the predictions between the self-consistent 1 ×10-5 approach and Voigt average (A ¼ I) are close to each other. Reasonably good agreement with

5 ×10-6 experiment measurement is observed near l ¼ 0:8 for the eﬀective piezoelectric, pyroelectric, and thermal expansion coeﬃcients, which suggests that 0 α 11 the ODF of poled barium titanate ceramics can be

The effective thermal expansion coefficients approximated by the Gaussian distribution func- -5× 10-6 tion with l 0:8. 0.01 0.1 1 10 100 ¼ µ Finally we would like to point out that for a pure elastic polycrystal, there are exact relationships Fig. 4. The effective thermal expansion coefficients for various between the effective thermal expansion coefficients textures; experiment data are from Dunn (1993a,b). and the effective elastic moduli (Schulgasser, 1987). We are unable to establish similar relationships for ferroelectric ceramics, except in some special situ- self-consistent approach and Voigt–Reuss aver- ations. For example, the exact relationships have ages all converge to the single crystal values, con- been established by Benveniste (1994) and Li et al. sistent with the exact relationship obtained by Li (1999) for piezoelectric polycrystals with fiber tex- et al. (1999). When l approach infinity, which ture, and by Chen (1994b) for certain specific implies that the ceramic is isotropic, the effective crystal symmetries. J.Y. Li / Mechanics of Materials 36 (2004) 949–958 957

7. Concluding remarks Dunn, M.L., 1995. Effects of grain shape anisotropy, property, and microcracks on the elastic and dielectric-constants of We have applied the self-consistent approach to polycrystalline piezoelectric ceramics. J. Appl. Phys. 78, 1533–1541. predict the effective pyroelectric and thermal Dunn, M.L., Taya, M., 1993. An analysis of piezoelectric expansion coefficients of ferroelectric ceramics, composite-materials containing ellipsoidal inhomogeneities. taking into account the texture change due to Proc. R. Soc. London A. 443, 265–287. domain switching during poling. The variations of Hashin, Z., 1984. Thermal expansion of polycrystalline aggre- thermal moduli with respect to the poling and gates. I. Exact results. J. Mech. Phys. Solids 32, 149– 157. texture have been demonstrated, and good agree- Jona, F., Shirane, G., 1993. Ferroelectric Crystals. Dover ments with theoretical results and experiment Publications, Inc., New York. measurements have been observed. Kreher, W.S., 1988. Internal stresses and relations between effective thermoelastic properties of stochastic solids––some exact solutions. Z. Angew. Math. Mech. 68, 147– 154. Acknowledgements Levin, V.M., 1967. On the coefficients of thermal expansion of heterogeneous materials. Mekhanika Tverdogo Tela 2, 88– Financial support by the Layman Award from 94. Research Council, University of Nebraska-Lin- Levin, V.M., Rakovskaja, M.I., Kreher, W.S., 1999. The effective thermoelectroelastic properties of microinhomoge- coln and NSF Nebraska EPSCoR Type II Grant is neous materials. Int. J. Solids Struct. 36, 2683–2705. gratefully acknowledged. Li, J.Y., 2000a. The effective electroelastic moduli of textured piezoelectric polycrystalline aggregates. J. Mech. Phys. Solids 48, 529–552. Li, J.Y., 2000b. Thermoelastic behavior of composites with References functionally graded interphase: a multi-inclusion model. Int. J. Solids Struct. 37, 5579–5597. Aboudi, J., 2001. Micromechanical analysis of fully coupled Li, J.Y., Bhattacharya, K., 2002. A mesoscopic electromechan- electro-magneto-thermo-elastic multiphase composites. ical theory of ferroelectric films ceramics. In: Fundamental Smart Mater. Struct. 10, 867–877. Physics of Ferroelectrics 2002, Proceedings of Workshop on Barnett, D.M., Lothe, J., 1975. Dislocations and line charges in Fundamental Physics of Ferroelectrics AIP Conference anisotropic piezoelectric insulators. Phys. Stat. Sol. B 67, Proceedings 626, pp. 224–231. 105–111. Li, J.Y., Dunn, M.L., 1998. Micromechanics of magnetoelec- Bhattacharya, K., Li, J.Y., 2001. Domain patterns, texture and troelastic composites: average fields and effective behavior. macroscopic electromechanical behavior of ferroelectrics. J. Intell. Mater. Syst. Struct. 9, 404–416. In: Proceedings of The 11th Williamsburg Workshop on Li, J.Y., Dunn, M.L., 1999. Analysis of microstructural fields in Fundamental Physics of Ferroelectrics, AIP Conference heterogeneous piezoelectric solids. Int. J. Eng. Sci. 37, 665– Proceedings 582, pp. 72–81. 685. Benveniste, Y., 1994. Exact Connections between polycrystal Li, J.Y., Dunn, M.L., Ledbetter, H., 1999. Thermoelectroelastic and crystal properties in 2-dimensional polycrystalline moduli of textured piezoelectric polycrystals: exact solutions aggregates. Proc. R. Soc. London A 447, 1–22. and bounds for film textures. J. Appl. Phys. 86, 4626–4634. Chen, T., 1994a. Micromechanical estimates of the overall Nan, C.W., Clarke, D.R., 1996. Piezoelectric moduli of thermoelectroelastic moduli of multiphase fibrous compos- piezoelectric ceramics. J. Am. Ceram. Soc. 79, 2563–2566. ites. Int. J. Solids Struct. 31, 3099–3111. Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall Chen, T., 1994b. Effective thermal expansion of piezoelectric Properties of Heterogenous Solids, second ed. Elsevier polycrystals. J. Mater. Sci. Lett. 13, 1175–1176. Science Publishers, North-Holland. Cross, E.L., 1993. Ferroelectric ceramics: tailoring properties Newnham, R.E., Skinner, D.P., Cross, L.E., 1978. Connectivity for specific applications. In: Setter, N., Colla, E.L. (Eds.), and piezoelectric–pyroelectric composites. Mater. Res. Bull. Ferroelectric Ceramics. Birkhauser€ Verlag, Berlin, pp. 1–85. 13, 525–536. Dunn, M.L., 1993a. Micromechanics of coupled electroelastic Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., composites––effective thermal-expansion and pyroelectric 1992. Numerical Recipes in FORTRAN, second ed. Cam- coefficients. J. Appl. Phys. 73, 5131–5140. bridge University Press, Cambridge, UK. Dunn, M.L., 1993b. Exact relations between the thermoelec- Qin, Q.H., Mai, Y.W., Yu, S.W., 1998. Effective moduli for troelastic moduli of heterogeneous materials. Proc. R. Soc. thermopiezoelectric materials with microcracks. Int. J. London A. 441, 549–557. Fract. 91, 359–371. 958 J.Y. Li / Mechanics of Materials 36 (2004) 949–958

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