Effective anti-plane properties of piezoelectric fibrous composites

Anti-plane shear of piezoelectric fibrous composites is theoretically investigated. The geometry of composites is described by the 2-dimensional geometry in a section perpendicular to the unidirectional fibers. The previous constructive results obtained for scalar conductivity problems are extended to piezoelectric anti-plane problems. First, the piezoelectric problem is written in the form of the vector-matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} -linear problem in a class of double periodic functions. In particular, application of the zeroth-order solution to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} -linear problem yields a vector-matrix extension of the famous Clausius–Mossotti approximation. The vector-matrix problem is decomposed into two scalar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document} -linear problems. This reduction allows us to directly apply all the known exact and approximate analytical results for scalar problems to establish high-order formulae for the effective piezoelectric constants. Special attention is paid to non-overlapping disks embedded in a two-dimensional background.


Introduction
We study piezoelectric anti-plane shear of piezoelectric fibrous composites in a stationary electromagnetic field when the electric charge induces the elastic stresses and deformations and vice versa. The geometry of the composites can be described by a two-dimensional geometry in a section perpendicular to the unidirectional fibers parallel to the x 3 -direction.
Different methods have been applied to estimate the effective constants of piezoelectric fibrous composites. The theory of homogenization of piezoelectric composites and exact formulae for the layered piezocomposites were described by Galka, Gambin, Telega and Wojnar (see the series of works [1][2][3][4][5][6][7]). Benveniste [8] derived a relation between the effective constants and a phase interchange connection for 2-phase piezoelectric fibrous composites in anti-plane statement. Milgrom and Shtrikman [9] studied the effective properties of a polycrystal made of uniaxial crystals. Schulgasser [10] established the exact relations for overall moduli of piezoelectric composites consisting of two and many transversely isotropic phases. Avellaneda and Swart [11] applied the effective medium theory and extended Hashin-Shtrikman bounds to isotropic polycrystals. They obtained expressions for the effective moduli in the effective medium approximations wherein each grain behaves like a sphere surrounded by a homogenized medium. Dunn and Taya [12] calculated the effective tensor of the piezoelectric 2-phase composites by application of various self-consistent theories. The effective medium approximations were developed out by Adler and Mityushev [13] in matrix form and applied to electrokinetic phenomena in porous media. The obtained results are valid for piezoelectric composites because of a common mathematical background. The application of self-consistent methods was extended by Levin et al. [14] to 2-dimensional piezoelectric composites. One of the methods [14], based on the effective field, gave formulae for the effective constants in explicit form; the other, the effective medium, gave implicit formulae.
Duality transformations were systematically applied in [13] to vector-matrix problems. The authors imposed conditions on the statistical properties of the local quantities in order to deduce exact explicit and implicit formulae when media are described by continuous local laws. These conditions and formulae are analogous to Matheron [15] and Dykhne [16] ones deduced for scalar problems. It is worth noting that formulae obtained in the framework of the 2-dimensional theory of duality transformations [13,15,16] are exact, and formulae deduced by self-consistent methods [11][12][13][14] are approximate and hold only for dilute and weakly inhomogeneous composites. The theory of the self-consistent methods was systematically discussed by Kanaun and Levin [17]; its limitations were analyzed in [18]. A method of integral equations for doubly periodic problems including anisotropic piezoelectric composites was developed in [19][20][21]. A rigorous mathematical study of the piezoelectric boundary value problems based on the theory of singular integral equations was presented in [22,23].
Many results are based on the method proposed by Rayleigh [24] and developed by McPhedran et al. [25][26][27][28] for regular lattices. Infinite systems of linear algebraic equations for the multipole coefficients of complex potentials were derived and truncated to get various low-order formulae of the effective conductivity. The method of Rayleigh was extended to elastic problems by Filshtinskii [20,29] and further by McPhedran et al. [30][31][32][33]. Mityushev et al. [34][35][36][37][38][39] extended the latter results for conductivity to arbitrary locations of disks per periodicity cell by a method of functional equations. High-order approximate analytical formulae for the effective conductivity were derived and exact formulae for regular arrays were deduced (see the survey [37]).
Guinovart et al. [40] (see also papers cited therein) extended the method of Rayleigh to piezoelectric fibrous materials. It was declared in [40] that closed-form analytical formulae are given for the effective properties. By closed-form analytical formulae, these authors mean the effective constants presented in a form that explicitly contains some parameters. These parameters can be numerically computed for fixed regular arrays by the truncation of infinite systems of linear algebraic equations following the method of Rayleigh. In the present paper, the terms exact formulae and approximate analytical formulae are used in a commonly accepted meaning. In particular, an effective constant σ e is given in such a form that only σ e is in the left part of the formula and the right part includes all the given geometrical and physical parameters explicitly in symbolic form, for instance, centers and radii of inclusions. Hence, contrary to [40], we have no any parameter in the right parts of our formulae that should be separately computed by a hidden numerical procedure.
This work is devoted to the extension of the constructive results [34][35][36][37][38][39] to piezoelectric anti-plane problems. First, the problem is written in the form of the vector-matrix R-linear problem [41,42] in a class of doubly periodic functions. Application of the zeroth-order approximation yields a vector-matrix extension of the famous Clausius-Mossotti approximation (67). This result is in agree with [13] and [14] obtained in another form. The vector-matrix problem is decomposed into two scalar R-linear problems. This reduction allows us to directly apply all the exact and approximate analytical formulae obtained before for scalar problems [30][31][32][33][34][35][36][37][38][39]. In particular, coupled anti-plane problems presented in [40] are reduced to scalar problems. This implies that truncated systems of order N 2 for numerical solution used in [40] are reduced to systems of order N by a simple linear transformation. It has to be noted that the direct application is performed only when the physical parameter defined by (74) is positive. In the case ≤ 0, an extension of the methods [30][31][32][33][34][35][36][37][38][39] has to be made.

Local equations and complex potentials
The interaction of the elastic and electric fields is local in the framework of the theory of continuum and can be modeled by local partial differential equations. Let E = (E 1 , E 2 , E 3 ) be the electric field strength, D = (D 1 , D 2 , D 3 ) the electric displacement vector, σ i j the stresses and u the elastic displacement. These values are considered as vector-functions of the spatial variables x j ( j = 1, 2, 3). Maxwell's equations in stationary electromagnetic problems become equations of electrostatics, where the free charge density vanishes. Anti-plane problems are stated for fibers parallel to the x 3 -axis when the elastic and electric forces do not depend on x 3 ; the stress tensor has only the nonzero components σ 13 and σ 23 , and the displacement has the form u = (0, 0, u 3 ).
Let a smooth oriented curve L locally divides the domains G + and G − on the plane (x 1 , x 2 ) occupied by different materials. The electric field strength in G + and G − is expressed through the electrostatic potential E(x 1 , x 2 ) = −∇φ(x 1 , x 2 ). The electric displacement vector and the stresses satisfy The combined effect of the anti-plane deformation and the electric fields in each section of the fiber composite is locally described by the coupled equations [19] σ i3 = c 44 where d 12 stands for the piezoelectric modulus, 11 the permittivity and c 44 the shear modulus.
These equations imply that the functions u 3 and φ are harmonic in the domains G + and G − . Therefore, they can be expressed in terms of the complex potentials where z = x 1 + i x 2 (i denotes the imaginary unit). The functions ϕ 1 (z) and ϕ 2 (z) are analytic in G ± and continuously differentiable in the closures of the considered domains. The following representations take place [19] where n = (n 1 , n 2 ) is the unit vector to AB, σ n3 = σ 13 n 1 + σ 23 n 2 , and D n = D 1 n 1 + D 2 n 2 .

Contact conditions and R-linear problem
Let the mechanical and electric contact between media which occupy the domains G + and G − be perfect. The perfect mechanical contact means that the normal limit stresses and the displacements from the both sides of L coincide: Here, n = (n 1 , n 2 ) is the outward normal vector to L. The ideal electric contact implies that the jump of the normal component of the electric displacement and the tangent component of the electric field strength vanish: where the tangent vector s = (−n 2 , n 1 ). The superscript "+" is assigned to all the values in the domain G + and "−" to G − . Using the complex potentials, we write (7) and (8) in the form Re ϕ + and where Equations (9), (11) and (12) are obtained from the first Eqs. (7) and (8) by integration along L [41]. Introduce the vector-functions the two real relations (10) and (12) become The conditions (9) and (11) can be written in the form where The two real vector-matrix conditions (15) and (16) can be cast in the complex vector form as The matrix D + is invertible since its determinant −(c 44 11 + d 2 12 ) is negative. It is convenient to introduce the vector-function and the normalized matrix Then, (18) takes the form of the following R-linear conjugation condition [41]: Introduce the vector-functions Differentiate the conditions (21) along the curve L on the tangent vector s. Then, (21) becomes (see analogous scalar manipulations in [41]) Here, the unit outward normal vector n = (n 1 , n 2 ) is presented as the complex value n(t) = n 1 + in 2 when t belongs to L. Equation (6) can be written in the vector-matrix form as where for definiteness it is assumed that the arc AB lies in G + .

Double periodic statement
We consider a 2-dimensional lattice Q defined by two fundamental translation vectors ω 1 and ω 2 expressed in terms of complex numbers. Let ω 1 > 0 and Im ω 2 > 0 as it is usually assumed in the theory of elliptic functions [43]. Let the zeroth cell Q (0,0) be the parallelogram determined by the vertices ± 1 2 ω j ( j = 1, 2). The lattice Q consists of the cells Q (m 1 ,m 2 ) = Q (0,0) + m 1 ω 1 + m 2 ω 2 , where m 1 and m 2 run over the set Z of integer numbers. Let mutually disjoint domains G k (k = 1, 2, . . ., N ) be located in Q (0,0) . The union of all the inclusions forms the non-connected domain We study the electrical and elastic fields in the doubly periodic composites when the domains G + and G − are occupied by piezoelectric materials whose properties are described by the matrices (17). It is convenient to introduce the complex potentials k (z) = (z) and where C j = (C 1 j , C 2 j ) T are constant vectors that model the external piezoelectric field applied to the composite in the following way. Let A and B from (6) have the complex coordinates z and z + ω j , respectively. Then, (25) yields It can be also written in the form where (24) is used. The constants P j can be expressed in terms of the average stresses and induction. First, introduce the average value of a doubly periodic function f (x 1 , x 2 ) over the periodicity cell Q (0,0) as the double integral where |Q (0,0) | = ω 1 Imω 2 denotes the area of the cell Q (0,0) . Periodicity of f (x 1 , x 2 ) implies that the integral (28) can be replaced by the integral over a rectangle Q defined by the vertices ± 1 2 ω 1 ± i 2 Imω 2 , Using (6), consider, for instance, the constant Integrate (30) by x 2 from − Imω 2 2 to Imω 2 2 and divide the result by Imω 2 . In accordance with (28), we obtain Similar arguments yield the following: Let the average values σ 3 and D ( = 1, 2) be known. Then, the constants P m ( , m = 1, 2) are calculated by (31)- (32). Thus, we arrive at the following. Boundary value problem: Given constants P j ( , j = 1, 2). To find the vector-function + (z) analytic in G + and continuously differentiable in the closure of G + , + (z) fulfills the quasiperiodicity conditions (25), where the constants C j satisfy (26). To find the vector-functions k (z) analytic in G k (k = 1, 2, . . . , N ) and continuously differentiable in G k ∪ ∂G k , the boundary values of these vector-functions satisfy the R-linear condition 1, 2, . . . , N ).
This boundary value problem can be considered as a vector-matrix generalization of the scalar problem discussed in [44]. This problem in the theory of composites provides the following standard method to compute the effective piezoelectric constants [19]. First, the averaged values σ 3 and D ( = 1, 2) are fixed. After, the constants P j are calculated with (31)-(32) and the problem (33) is solved in a class of quasiperiodic functions. Here, the jumps C j from (25) satisfy the conditions (26). Then, the averaged deformations ∂u 3 ∂ x and electric fields E ( = 1, 2) are computed as linear combinations of σ 3 and D . This linear dependence determines the effective piezoelectric tensor P. More precisely, few independent problems should be solved to determine all the components of P. For instance, two problems should be solved for macroscopically isotropic composites.
We propose another method to determine P, which is convenient in symbolic computations. Let the host medium and inclusions be made from the same material. This implies that D + = D − and R = 0 by (20). Then, the R-linear condition (33) becomes the condition of analytic continuation of 0 (z) into G k (k = 1, 2, . . . , N ). Therefore, this degenerate case corresponds to the problem when a quasi-periodic vector-function 0 (z) analytic in Q (0,0) has to be found. The components of 0 (z) have to be linear functions az + b. It is convenient to take two linearly independent solutions of the degenerate problem, Then, (25) yields the following: Now, we come back to the general condition (33) and solve the problem (33), (25) with the prescribed jumps (35). After, we calculate the averaged fields and symbolically compute P by the averaged piezoelectric law.

Effective elastic constants for piezoelectric problems
This section is devoted to the calculation of the effective piezoelectric tensor P by the method outlined at the end of the previous section. For simplicity, it is assumed that the inclusions G k are distributed in such a way that the considered composite is isotropic in macroscale in the plane perpendicular to the fibers. The average equations (3) for macroscopically isotropic composites take the form Equation (36) determines the effective piezoelectric tensor P. For macroscopically isotropic composites, P can be presented by the matrix The signs in the second column correspond to Eq. (36). They are chosen for convenience of the further calculations. Moreover, it makes the tensor P symmetric. Equation (36) can be written in vector-matrix form as In order to find the components of P, we calculate two sets of the average values from (36) corresponding to two different external fields for i = 1. First, using the representations (4) and the definition (14) of the vector-functions k (z), we calculate the integral as follows: The vector-functions k (z) and k (z) are related by the equation (see (19)) In order to transform (39), we use Green's formula as follows: where z = x 1 + i x 2 . The double integrals from (39) can be reduced to the linear integrals as and where t = x 1 + i x 2 ∈ ∂G k . The integral on ∂ Q (0,0) from (42) can be calculated through the increments (25) Formula (39) becomes Application of (15), (40) and (20) to the latter integrand yields Then, (45) becomes where Application of similar arguments yields the following: Formulae (38), (47) and (49) produce formulae for the effective tensor P in terms of the integrals (48) in the following way. Substitute two vectors (35) into (47) and (49) instead of C 1 and C 2 . Then, we obtain two vectors and Re C The corresponding local fields can be averaged and Eq. (38) applied to these fields implies that where where I is the identity matrix. The elements of V are given by (47) with the constant vectors C 1 and C 2 given by (35). It follows from (52) that Consider two R-linear problems (33) stated at the end of the Sect. 4 ( ) with the quasi-periodicity conditions (25) when the jumps of ( ) 0 (z) are given by (35) (1) Here, is the number of the R-linear problem ( = 1, 2); k is the number of the inclusion; and j is the number of the jump condition in each th problem. Introduce the vector-functions and the matrices Differentiation of (55) on a tangent parameter of ∂G k yields [41] ( ) where n(t) is the normal outward vector to ∂G k expressed in terms of complex values. Differentiation of (56) implies double periodicity of ( ) 0 (z). Using (48) and (41), we introduce the matrix Then, (54) can be written in the extended form as This formula (61) can be considered as an extension of Mityushev's formula [34,37,44] to piezoelectric fiber composites.
Example 1 Let mutually disjoint disks G k = {z ∈ C : |z − a k | < r } (k = 1, 2, . . . , n) be located in Q (0,0) . The normal vector to the circle |t − a k | = r has the form n(t) = 1 r (t − a k ). Then, (59) becomes ( ) 1, 2, . . . , n). (62) The integral (60) can be calculated by the mean value theorem for harmonic functions Let ν denote the concentration of disks in the periodicity cell Substitution of (63) into (61) yields the following: For dilute and weakly inhomogeneous composites [18], the vector-functions ( ) k (z) satisfying (62) are approximated by constant vectors obtained by differentiation of (34). Hence, the matrices k (z) are approximated by the unit matrix: Substitution of these approximations into (65) and application of the simple rational approximation in ν yield the following: The latter formula can be considered as an extension of the famous Clausius-Mossotti approximation to piezoelectric fiber composites. Direct computations show that the matrix in the right part of (67) is symmetric.

High-order approximate analytical formulae
The R-linear vector-matrix problem (33) is the key of the high-order approximate analytical formulae for the effective piezoelectric tensor P determined by (61). Scalar R-linear problems and their application to the effective conductivity tensor were discussed in [34,37,44]. In order to use these constructive results for the scalar problem, one can try to decompose the vector-matrix problem (33) into two scalar ones. The matrix R is defined by (20) through two positively determined matrices D + and D − . Let it be presented in the form where T is a non-singular matrix, and is diagonal Introduce the contrast parameters and the constants Then, the matrix R can be written in the form It is assumed that the matrix R is not singular; hence, Let the constant be positive. In this case, the matrix T = {t m } is real and has the form The inverse matrix T −1 = {t * m } is given by The real elements of the matrix (69) read as follows: Hence, the similarity relation (68) between R and holds. The eigenvalues (77) always satisfy the inequality In order to prove this, we consider the equivalent relation Further, consider the case h(ρ 1 + ρ 2 ) + 2ρ 3 ≥ 0. In the opposite case, the proof is similar. It is sufficient to demonstrate that h(ρ 1 + ρ 2 ) + 2ρ 3 + √ ≤ 2(h + 1). The latter inequality for positive given by (74) is equivalent to Use of simple transformations yields the following: This inequality follows the form Therefore, (78) is proved. Substitute the representation (68) into (59) and multiply the result by T −1 1, 2, . . . , n; = 1, 2).
Then, (82) becomes ω ( ) since the matrix T is real. The vector-matrix problem (85) can be decoupled onto the scalar problems where ω (m, ) k (t) is the mth coordinate of the vector ω ( ) k (t) (or the element (m, ) of the matrix ω k (t)). The method [34,37,44] can be applied to the scalar problems (86) since (78) holds. Following Example 1, consider non-overlapping circular inclusions. Then, (86) becomes The effective tensor P has the form (65) where by (83)-(84) In the zeroth approximation, k (z) ≈ I (see (66)). Then, the zeroth approximation for ω k (z) is determined by (83) Therefore, we arrive at four scalar problems (86) ( , m = 1, 2) with the zeroth approximations where t * m denote the elements of the matrix T −1 . The scalar problem (86) and (91) is formally coincided to the conductivity problem with the contrast parameter λ m and the external flux expressed by the vector t * m (1, 0) T . The problems (87) and (91) were solved in [34,37,44] for arbitrary locations of non-overlapping inclusions |z − a k | < r (k = 1, 2, . . . , n) in the periodicity cell. For macroscopically isotropic composites, the ratio σ e of the effective conductivity to the conductivity of the host material was written in the form [34,37] where the solution of the problem (87) and P(λ m ) is related by the equation The function P(λ m ) can be explicitly written in terms of the series introduced by Mityushev [36] (called by him the generalized Eisenstein-Rayleigh series). Substitution of (93) into (89) yields .

N. Rylko
Application of the explicit formulae (75)-(76) to the elements ψ m of the matrix (88) yields Formula (65) can be written in the form Symbolic computations of P(λ) for various locations of the disks by the method [34,37,44] and substitution of the results into (94)-(95) yield analytical formulae for the effective tensor P. For instance, P(λ) can be approximated by [45] where Here, E p (z) denotes the Eisenstein function of order p. An exact formula for the effective conductivity of regular arrays of disks, hence for P(λ), is written in [37].
Example 2 Consider a random composite when the disks obey the uniform non-overlapping distribution. Then, P(λ) can be approximated up to O(ν 6 ) by the following expression [39]: It is shown in [39] that the precision O(ν 5 ) in (98) is sufficient to give excellent results for |λ| < 1. For |λ| close to unity, it is better to use Padé approximations [39].
Example 3 Consider equal ellipses located in the square lattice formed by ω 1 = 2 and ω 2 = 2i in such a way that one ellipse lies exactly in one cell. Let the angle between the major semi-axis and the axis O X 1 be a random uniformly distributed variable. Then, P(λ) can be approximated up to O(ν 2 ) by the following formula [38]: where q denotes the ratio of the semi-axes of ellipses. The tensor P −1 was computed in [21] for tetragonal array of ellipses whose axes were parallel to the coordinate axes. Let the matrix be made of PZT-4 piezoceramics with the parameters c 44 = 2.56 × 10 10 N/m 2 , 11 / 0 = 729 ( 0 stands for the vacuum permittivity) and e 15 = 12.7 C/m 2 ; the fibers of PP-2 with the parameters c 44 = 10 × 10 10 N/m 2 , 11 / 0 = 500 and e 15 = 0 C/m 2 . The numerical results of [21] were presented for q = 0.5. Consider a similar numerical example with the same physical properties and the same ellipses, but randomly oriented in the square array. In this case, Eq. (77) gives the real eigenvalues λ 1 = −0.186 and λ 2 = 0.592. Computations performed with formulae (95) Fig. 2 depends rather linearly on ν. Example 4 Checkerboard 2-dimensional composites can be considered as a limit case of the square inclusions. Application of Dykhne's formula [16] to (92) with ν = 1 2 yields General exact formulae for vector-matrix problems obtained in the framework of the theory of duality transformation can be found in [13].

Discussion
The present work is devoted to the extension of the constructive results [34][35][36][37][38][39] to piezoelectric anti-plane problems by the use of the vector-matrix R-linear problem [41,42] in a class of doubly periodic functions. The vector-matrix extension (67) of the Clausius-Mossotti approximation is obtained (cf. [13] and [14]). The key point to get high-order approximations formulae for the effective constants is the decomposition of the vector-matrix problem into scalar R-linear problems. Further, the known exact and approximate analytical formulae obtained before for scalar problems [30][31][32][33][34][35][36][37][38][39] are directly applied. This direct application is described when the physical parameter introduced by (74) in Sect. 6 is positive. We now discuss the physical restrictions on and the case ≤ 0. The dimensionless contrast parameters (70) satisfy the inequality −1 ≤ ρ j ≤ 1 ( j = 1, 2, 3). The dimensionless constants c and e from (71) are always negative and h is positive. It follows from accessible data for the piezoelectric materials [21,22] that from (74) can be positive, negative and equal to zero. Consider below these cases and simple examples to demonstrate that all these cases take place.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.