Analytical expressions to estimate the effective piezoelectric tensor of a textured polycrystal for any crystal symmetry

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier In Mechanics of Materials, Vol. 151, on December 2020, available at: https://doi.org/10.1016/j.mechmat.2020.103604 Copyright 2020 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Analytical expressions to estimate the effective piezoelectric tensor of a textured polycrystal for any crystal symmetry Julieta L. Buroni Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Deán Funes 3350, B7602AYL, Mar del Plata, Argentina Federico C. Buroni E-mail: [email protected] Department of Mechanical Engineering and Manufacturing, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Seville, Spain Adrián P. Cisilino INTEMA & Department of Mechanical Engineering, Universidad Nacional de Mar del Plata & CONICET, Av. Juan B. Justo 4302, B7608FDQ Mar del Plata, Argentina Roderick Melnik MS2 Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, 75 University Ave W, Waterloo, Ontario, Canada N2L 3C5 Luis Rodríquez-Tembleque and Andrés Sáez Department of Continuum Mechanics and Structural Analysis, Universidad de Sevilla, Camino de los Descubrimientos s/n, Seville E-41092, Spain Abstract. This work introduces a set of analytical expressions that simplifies the procedure of obtaining closed-form formulas to estimate the effective piezoelectric properties of polycrystalline aggregates formed by crystals of all the 21 noncentrosymmetrical classes and with arbitrary textures. The expressions are derived from orientational averages of third-order piezoelectric tensors of the individual crystals that are weighted by orientational distribution functions. The averaging is done by using generalized spherical harmonic series expansions. Improvements with respect to previous works in the literature are twofold: all crystal symmetries are considered and no symmetry restrictions are imposed to texture. The versatility of the introduced expressions is demonstrated on an example of BaTiO3polycrystals with uniaxial and biaxial textures characterized by single and double Gaussian distributions, respectively. The results for uniaxial texture are in perfect agreement with the results published in the literature. The biaxial texture is discussed in detail and analyzed for a set of limiting cases. Manuscript File Click here to view linked References Averaging piezoelectric third-order tensors 2 1. Introduction Piezoelectric ceramics (Uchino, 2017) are generally used in the form of a collection of perfectly bonded piezoelectric crystals with a large number of distinct polarization orientations. Macroscopic piezoelectric properties of the material are governed by this microscopic inhomogeneous crystal morphology. When the material is an aggregate of randomly oriented piezoelectric crystals, no net polarization is realized and so the material is not macroscopically piezoelectric. To achieve macroscopic piezoelectric properties, the individual crystals must have a preferential orientation. Generally, piezoelectric properties exhibit large anisotropy, which is an attractive feature for material design. Preferred orientation of the crystallographic domains can be accomplished by controlling growth parameters during fabrication (Kim et al., 2006) or by subjecting the material to a large electric field at high temperature (Li et al., 2005). Optimization methods can be used to design materials specific piezoelectric properties by tailoring their orientation distribution (texture). In the framework of the implementation of an optimization algorithm, Finite Element Analysis (FEA) is a versatile tool to compute effective piezoelectric properties (the optimization objective function) by means of a numerical homogenization approach, see for instance Jayachandran et al. (2011). However, closed-form analytical expressions for effective piezoelectric properties are always preferable over the FEA homogenization approach, as they simplify implementations and speed up calculations. Effective piezoelectric properties can be estimated by averaging the properties of single crystals in the aggregate taking into account macroscopic texture. To calculate the effective properties for a polycrystal, the best averaging method is the self-consistent, extended to piezoelectric polycrystals by Li (2000). However this method requires computation of piezoelectric Eshelby tensor which involves, in general, numerical integration. On the other hand, there are the simple volume averaging approaches, like the Voigt model (see Li and Dunn (2001)), which are attractive because they allow to deduce closed-form expressions for the effective properties. In particular, the Voigt model assumes uniform strain and electric fields in the crystallites. This assumption is valid for piezoelectric polycrystals with fiber textures under certain conditions (see Li et al. (1999)) for which exact estimates of some of the electroelastic moduli result. Working in the context of fiber texture, Li et al. (1999) and Li (2000) computed Voigt-Reuss estimations for the electroelastic moduli of polycrystals by approaching the problem with generalized spherical harmonics. Specifically, Li et al. (1999) considered fiber texture of orthorhombic single crystals belonging to class symmetry 2mm, which includes as particular cases classes of tetragonal 4mm and hexagonal 6mm symmetries (see Nye’s book (Nye, 1985)). An interesting finding by Li (2000) was that aggregates of 4mm crystals of barium titanate (BaTiO3) with their orientations around the polarization direction characterized by a Gaussian distribution present an amplification effect of the piezoelectric coupling with respect to the single crystal properties. Recently, some of the Averaging piezoelectric third-order tensors 3 authors of the present work have used the Li’s framework for modelling modern lead-free piezocomposites to find interesting behaviours attributable to such amplification effect of the polycrystalline phase (Krishnaswamy et al., 2019a,b, 2020a,b,c). Previous works address only orthorhombic crystal symmetry with transversely isotropic texture. This work presents a comprehensive approach to derive closedform expressions for the estimation of the effective piezoelectric stress tensor of crystal aggregates of any symmetry and general texture. The proposed approach is based on the previous works by Li et al. in the use of generalized spherical harmonics expansion and extends the ideas by Bunge (1982) and Roe (1965) among others, from elasticity to piezoelectricity. The derived analytical expressions are verified and their versatility demonstrated through a study problem consisting in a polycrystalline aggregate of BaTiO3with various orientational distribution functions. It is shown how the general expressions introduced in this work can be reduced to those results by Li (2000) for specific cases. The repeated index convention for addition is used in this work. Explicit sums are indicated in expressions when deemed convenient for the sake of clarity. 2. Preliminaries We concern with the computation of the anisotropic electromechanical coupling of polycrystal piezoceramics from the properties of single crystallites and their orientations. A simple approximation to compute this coupling is through the volume average of the third-order piezoelectric stress tensor eijk, which satisfies eijk =eikj (i, j, k =1...3). Since in this case eijk depends only on the crystal orientation g(xi)(i=1...3) at each point x, the integration over a representative volume element Vcan be carried out in two steps, as usual. We first integrate over all those volume elements dVwith orientation g, and then over all orientations g, thereby the average tensor is computed by hei=8⇡2Ie(g)w(g)dg, (1) where the function w:SO(3) !R0–called Orientation Distribution Function (ODF)– appears as a weight function that accounts for the volumetric density of crystallites oriented in dg. This Haar measure dgis —after normalization— given by dg=1/(8⇡2)sin✓d✓d dwhere ( ,✓,)are the Euler’s angles. Then, the ODF is normalized such as 8⇡2Iw(g)dg=1.(2) Therefore, by expressing the crystal orientation in terms of ( ,⇠=cos✓,), the Voigt average becomes hei=Z2⇡ 0Z2⇡ 0Z+1 1 e(⇠, ,)w(⇠, ,)d⇠d d,(3) Averaging piezoelectric third-order tensors 4 Note that the normalization factor 8⇡2above results in a isotropic ODF, wiso =1/(8⇡2) (Bunge, 1982; Roe, 1965). Among the different alternatives, we adopt the following convention to specify rotations through Euler’s angles: one begins with the principal material coordinate system x0 i(i=1,2,3) fixed to the crystal and with the axes being parallel to those of the global coordinate system xi(i=1,2,3); then x0 i(i=1,2,3) is first rotated about the x0 3-axis through the angle ; the second rotation is about the x0 2-axis (in its new orientation) through ✓and, finally, the third rotation is again about the x0 3-axis (in its new orientation) through the angle . All positive rotations are in counterclockwise direction. The piezoelectric components of e0fixed to a crystal can be expressed in the global coordinate system xi(i=1,2,3) according to the transformation law for third-order tensors eijk(⇠, ,)=⌦ im⌦jn⌦koe0 mno =Timjnkoe0 mno,(4) where ⌦–the matrix representation of the orientation gin the Euler’s space– is the orthogonal transformation matrix (SO(3)) given by ⌦=0 B @ cos(✓)cos( )cos()sin( )sin()cos(✓)cos( )sin()sin( )cos()sin(✓)cos( ) cos(✓)sin( )cos()+cos( )sin()cos( )cos()cos(✓)sin( )sin()sin(✓)sin( ) sin(✓)cos()sin(✓)sin()cos(✓) 1 C A.(5) The sixth-order matrix Timjnko condenses the triple tensor product ⌦im⌦jn⌦ko; it possesses major symmetries only (minor symmetries are not present because ⌦is nonsymmetric) given by the permutations of the disjoint cycles of indices Cycles[(3 5)(4 6)],Cycles[(1 3)(2 4)],Cycles[(1 3 5)(2 4 6)].(6) These symmetries allow a considerable reduction in the number of components of Timjnko to be computed. Thus, of the 729 components of Timjnko only 249 are different. Finally, the averaging of equation (4) results in heijk(⇠, ,)i=hTimjnkoe0 mnoi=hTimjnkoie0 mno,(7) which shows that the estimation of the effective piezoelectric tensor of the polycrystalline aggregate reduces to finding the hTimjnkoifor the corresponding ODF. In what follows, the prime symbol will be dropped from the piezoelectric constants in the crystal coordinate system in order to allow a cleaner notation. 3. Averaging in terms of generalized spherical harmonic series expansions Following Gel’fand et al. (1963), generalized spherical harmonics form a complete orthogonal basis for the Hilbert space L2(SO(3)), which is the set of all square integrable real-valued functions on SO(3) with inner product defined by (H1,H 2)=IH1(g)H⇤ 2(g)dg, H1,H 22L2(SO(3)),(8) Averaging piezoelectric third-order tensors 5 where ⇤denotes the complex conjugate. Thus, a general real function H2L2(SO(3)) can be expanded in series of generalized spherical harmonics. This is a useful approach to evaluate the average of a general real function hHi. To do this, we expand both the ODF w(⇠, ,)and H(⇠, ,)into generalized spherical harmonics series (Roe, 1965) as follows: w(⇠, ,)= 1 X l=0 l X m=l l X n=l WlmnZlmn(⇠)eim ein,(9) and H(⇠, ,)= 1 X l=0 l X m=l l X n=l HlmnZlmn(⇠)eim ein,(10) where i=p1and Zlmn(⇠)is a generalization of the Legendre associated function Pmn l that can be expressed as Zlmn(⇠)=i nmr2l+1 2Pmn l(⇠),(11) being Pmn l(⇠)= (1)lminm 2l(lm)! (lm)!(l+n)! (l+m)!(ln)!1 2 ⇥(1 ⇠) (nm) 2 (1 + ⇠)(n+m) 2 dln d⇠ln⇥(1 ⇠)lm(1 + ⇠)l+m⇤ .(12) The associated Legendre function Pmn lcan be either real or purely imaginary, according to whether m+nis even or odd, respectively. Consequently, Zlmn(⇠)is always realvalued. The coefficients of the expansions in (9) and (10) are (Bunge, 1982; Kocks et al., 2005) Wlmn =1 4⇡2Z2⇡ 0Z2⇡ 0Z+1 1 w(⇠, ,)Zlmn(⇠)eim eind⇠d d,(13) the so-called texture coefficients, and Hlmn =1 4⇡2Z2⇡ 0Z2⇡ 0Z+1 1 H(⇠, ,)Zlmn(⇠)eim eind⇠d d.(14) Note that all information about the ODF is contained in the texture coefficients, which are complex quantities satisfying Wlmn =(1)m+nW⇤ l¯m¯n,(15) due to the symmetry properties of Zlmn(⇠)(Roe, 1965). In equation (15), ¯m=m. From the normalization condition (2), it can be shown that W000 =1 4p2⇡2.(16) In practice, the normalization of w(⇠, ,)is prefered by ensuring (16) instead of (2). Averaging piezoelectric third-order tensors 6 Expansion coefficients Hlmn are complex and typically involve integration of simple trigonometric functions (Hcontains products of direction cosines) and the generalized spherical harmonics. Their analytical expressions can be found once His specified. Finally, the expansion for the average of a general real function hHi,is (Bunge, 1982; Kocks et al., 2005) hHi=4⇡2 R X l=0 l X m=l l X n=l HlmnWlmn,(17) where, from the truncation theorem by Ferrari and Johnson (1988), R=3, the tensor rank of eijk. In fact, all coefficients Hlmn are zero for l>3. It is worth to note that, unlike the works by Li (2000); Krishnaswamy et al. (2019a); Li et al. (1999), the present analysis does not imposed any restriction on Wlmn. In the next, we will use (17) to obtain the expressions for the average rotation tensor in (7), which, in turn, will be used to compute the effective piezolectric tensor. 4. Effective piezoelectric tensors for all crystal symmetries Note first that each of the components Timjnko are real functions Timjnko :SO(3) !R, similar to function H. Then, derivation of the expressions for the average rotation matrix in (7) can be performed via expression (17). This methodology involves the computation of the coefficients Hlmn for each of thse 249 different components of Timjnko according to (14). This was done analytically using Mathematica (2020). Due to space limitations, the full expressions are reported in Supplementary Material 1. All 249 components are necessary in order to consider a general case with any symmetry. Two sources of symmetry are considered when computing the average piezoelectric tensor (7): the ones due to the crystallite structures and the others due to the ODF. No symmetry restrictions are imposed on the ODF –and hence to the texture coefficients in (17)– in this work. In respect to the crystallite symmetries, all the non-centrosymmetrical classes (Nye, 1985) are considered. The expressions of (7) for each of the 21 classes are presented in Supplementary Material 2. The heijki in Supplementary Material 2 are functions of the texture coefficients Wlmn, which are necessary to specify for each case of analysis according to the adopted ODF. The property (15) has been used to present the expressions of heijkiin real form. Note that from the 21 non-centrosymmetrical classes only 20 are piezoelectric. Thus, the cubic class 432 results in all components of heijkiidentically zero, no matter what the Wlmn are. Some of the piezoelectric classes share the matrix structure reducing the amount of expressions as it can be observed in Supplementary Material 2. Two supplementary materials provide all the details of the main contribution of this work. The next section analyzes the tetragonal 4mm class in detail. Averaging piezoelectric third-order tensors 7 5. Study problem: Tetragonal class 4mm The theory introduced above is applied for the analysis of polycrystalline aggregate of BaTiO3in order to verify and illustrate the derived results. Barium titanate is both a lead-free material (Saito et al., 2004) and environmentally friendly in its processing (Ibn-Mohammed et al., 2017). Interesting novel applications (Wang et al. (2018) among others) have been proposed while BaTiO3-based composites can be fabricated in a scalable manner using emerging additive manufacturing methods (Kim et al., 2017, 2019; Phatharapeetranun et al., 2017). Li (2000), among others (see for example (Li et al., 1999; Ruglovsky et al., 2006)), has studied the average piezoelectric properties of BaTiO3polycrystalline aggregates. The BaTiO3crystal belongs to the point group class 4mm, with their piezoelectric constants listed in Table 1 (Li, 2000). Thus, the expressions for the average components heijmifor the tetragonal symmetry are retrieved from Supplementary Material 2 and summarized below: he111i=4 105⇡2⇣p14(5eAeB)<⇣p3W310 p5W330⌘+ 14p3eB<(W110)⌘, (18) he122i=4 315⇡2⇣14p3<(W110)(2eAeB+3e C)+ p14(eB5eA)<⇣p3W310 +3 p5W330⌘⌘, (19) he133i=8⇡27<(W110)(2eAeB+3e C)+2 p14(5eAeB)<(W310) 105p3,(20) he123i=8⇡2(5eAeB)=(W320) 3p105 ,(21) he113i=2 315⇡2⇣7p6<(W100)(2(eA+e B)+3e C)+ 2p7(5eAeB)<⇣3p2W300 2p15W320⌘⌘, (22) he112i=4 315⇡2⇣7p3=(W110)(2(eA+e B)+3e C)+ p14(5eAeB)=⇣p3W310 3p5W330⌘⌘, (23) he211i=4 315⇡2⇣14p3=(W110)(2eAeB+3e C)+ p14(eB5eA)=⇣p3W310 3p5W330⌘⌘, (24) he222i=4 105⇡2⇣p14(5eAeB)=⇣p3W310 +p5W330⌘+ 14p3eB=(W110)⌘, (25) Averaging piezoelectric third-order tensors 8 he233i=8 105p3⇡2(7=(W110)(2eA+e B3eC)+ 2p14(eB5eA)=(W310)⌘, (26) he223i=2 315⇡2⇣7p6<(W100)(2(eA+e B)+3e C)+ 2p7(5eAeB)<⇣3p2W300 +2 p15W320⌘⌘, (27) he213i=8⇡2(5eAeB)=(W320) 3p105 ,(28) he212i=4 315⇡2⇣7p3<(W110)(2(eA+e B)+3e C)+ p14(5eAeB)<⇣p3W310 +3 p5W330⌘⌘, (29) he311i=4 315⇡2⇣7p6<(W100)(2eAeB+3e C)+ p7(eB5eA)<⇣3p2W300 2p15W320⌘⌘, (30) he322i=4 315⇡2⇣7p6<(W100)(2eAeB+3e C)+ p7(eB5eA)<⇣3p2W300 +2 p15W320⌘⌘, (31) he333i=4 105 ⇣2p14⇡2(eB5eA)<(W300)+7 p6⇡2eB<(W100)⌘,(32) he323i=4 105p3⇡2(7=(W110)(2(eA+e B)+3e C)+ 4p14(eB5eA)=(W310)⌘, (33) he313i=4 105p3⇡2(7<(W110)(2(eA+e B)+3e C)+ 4p14(eB5eA)<(W310)⌘, (34) and he312i=8⇡2(5eAeB)=(W320) 3p105 ,(35) where eA=2e113 +e311,eB=4e113 +2e311 +3e333 and eC=2e113 4e311. Note that this set of analytical expressions reduce to those presented by Li (2000) when ODF is such that texture coefficient W320 vanishes. However, expressions (18) - (35) are more general as shown below. To verify these results, we first consider the ODF w(⇠, ,)=(⇠1),(36) where (⇠1) is the Dirac delta function at ⇠=1. This ODF describes an aggregate with all the crystallites oriented with ✓=0and random values for and , which Averaging piezoelectric third-order tensors 15 Figure 7. Effective piezoelectric moduli of polycrystal BaTiO3as a function of the Gaussian distribution parameter ✓for µ✓=0,µ =⇡/3and  =1. Figure 8. Effective piezoelectric constant he112iof the BaTiO3polycrystal as a function of µ and  for µ✓=0and ✓=0.6. Table 2. Matrix structures of the effective piezoelectric coupling tensors resulting from the two Gaussian ODFs. ODF Crystal matrix structure Polycrystal matrix structure wA0 @ 0000e223 0 000e223 00 e311 e311 e333 000 1 A0 @ 0000he223i0 000he223i00 he311ihe311ihe333i000 1 A wB0 @ 0000e223 0 000e223 00 e311 e311 e333 000 1 A0 @ he111ihe122ihe133ihe123ihe113ihe112i he211ihe222ihe233ihe223ihe123ihe212i he311ihe322ihe333ihe323ihe313ihe123i 1 A Averaging piezoelectric third-order tensors 16 6. Conclusions This work provides general analytical expressions for volume averages of piezoelectric properties of polycrystalline aggregates. The expressions are derived from orientational averages of third-order piezoelectric tensors that are weighted by orientational distribution functions that account for texture. The expressions are derived by using spherical harmonic series expansions. Improvements of this work with respect to the previous works in the literature are twofold: all crystal symmetries are considered and no symmetry restrictions are imposed to texture. Thus, this work provides the analytical formulas to compute the average piezoelectric tensor of polycrystalline aggregates formed by crystals of all the 21 noncentrosymmetrical classes. These expressions are open to be specialized for any texture, which is specified in terms of Euler´s angles ( ,✓,). The versatility of the introduced expressions is demonstrated for BaTiO3 polycrystals with uniaxial texture given by a single Gaussian dispersion for ✓, and with biaxial texture given by a double Gaussian dispersion for ✓and . The results for the first case, which consisted in a Gaussian distribution centered around ✓=0, showed to be in perfect agreement with results published in the literature. This case has the particularity that all the texture coefficients and all the expressions of the average piezoelectric properties are real. On the other hand, no previous study was found to verify the results for the simple Gaussian distributions with mean value ✓6=0 and for the case with biaxial textures. The biaxial texture leads to complex-valued texture coefficients and complex formulations for the averaged piezoelectric properties. Results for the BaTiO3polycrystal with biaxial texture have been discussed in detail and analyzed for a set of limiting cases. In all cases, the results showed to be consistent. The main contribution of this work is the set of analytical expressions that simplifies obtaining closed-form formulas to estimate the effective piezoelectric properties of polycrystals with arbitrary symmetries and textures. These expressions are suitable to perform parametric studies like those presented in this work for BaTiO3polycrystals, and –provided the textures characterized by smooth ODFs– to implement efficient analytical optimization methods to find configurations at the microstructural level in order to maximize piezoelectricity. It is straightforward to adapt the expression herein derived for the piezoelectric stress tensor eto other piezoelectric properties, such as d,g,h(see IEEE Standard on Piezoelectricity (1988)) or the piezomagnetic moduli. Author’s contributions JLB: Investigation, Software, Validation, FCB: Conceptualization, Formal Analysis, Methodology, Visualization, Project Administration, Writing - Original draft preparation, Review & Editing, APC: Supervision, Resources, Writing - Original draft preparation, Review & Editing, RM: Writing – Review & Editing, LRT: Project REFERENCES 17 Administration, Writing - Review, AS: Writing - Review Acknowledgments This work was supported by the Ministerio de Economía y Competitividad of Spain under project DPI2017-89162-R, the Consejería de Economía, Conocimiento, Empresas y Universidad of the Junta de Andalucía (Spain) under project P18-RT-3128 and the Universidad Nacional de Mar del Plata (Argentina) under project 15/G511. R.M. is also acknowledging support of NSERC and CRC Program. Data Availability The data that supports the findings of this study are available within the article [and its supplementary material]. References Bunge, H. 1982. 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