Analytical and semi-analytical modeling of effective moduli bounds: Application to transversely isotropic piezoelectric materials

Abstract

In this article, analytical and semi-analytical models of upper and lower bounds for the effective moduli of transversely isotropic piezoelectric heterogeneous materials based on the generalized Hashin–Shtrikman variational principle are presented. Compact matrix formulations are used to derive closed-form bound expressions for coupled and uncoupled effective moduli. Analytical models are given for some uncoupled coefficients and simplified formulations for the others. For more narrow bounds, downstream and upstream bounds are developed based on an incremental procedure. Numerical predictions are performed based on the developed methodological approaches, and the obtained results showed the applicability and effectiveness of the proposed models for transversely isotropic elastic and piezoelectric composite materials with ellipsoidal reinforcements of different types and shapes.

Keywords

micromechanics piezoelectric analytical bounds downstream and upstream bounds Hashin–Shtrikman bounds effective electroelastic moduli transverse isotropic material ellipsoidal inclusion

Introduction

Micromechanical models represent an efficient tool to evaluate the overall properties of heterogeneous bodies. Several methods, mostly based on the fundamental solution of Eshelby (1957) of the problem of an ellipsoidal inclusion in an infinite isotropic medium, have been proposed to compute estimation or bounds of the effective elastic or electroelastic properties of composite materials when the volume fractions or the actual geometry of the constituent phases is known (Bakkali et al., 2013; Dunn and Taya, 1993a-1993b, El Ouafi et al., 2015; Fakri and Azrar, 2010; Fakri et al., 2003; Mura, 1982). Viscoelectroelastic properties for multi-coated piezoelectric composites are investigated by Azrar et al. (2014).

The classical variational principle was established by Hashin and Shtrikman (1962, 1963) and has been applied to derive upper and lower bounds for the effective elastic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. Bounds of multiphase isotropic and homogeneous composite with arbitrary geometry are given by Walpole (1966a,1966b). The Hashin–Shtrikman variational formulations have been developed in order to obtain tighter bounds by taking into account the geometry of the constituent phases (Brisard et al., 2010; Hori and Nemat Nasser, 1998; Milton and Kohn, 1988; Nemat-Nasser et al., 1993).Cherkaev (2014) obtained optimal design for metamaterial composites. Based on the variational principle and the equivalent inclusion method, the linear work-hardening behaviour of some piezoelectric composites is investigated by Huang (1996). Bisegna and Luciano (1996,1997) obtained variational bounds for the overall properties of piezoelectric composites and of periodic heterogeneous media with non-linear and non-local piezoelectric constitutive relationships. Rodríguez-Ramos et al. (2004) used the Hashin–Shtrikman variational principles to establish bounds of non-linear piezoelectric materials. Variational bounds for the effective moduli of heterogeneous piezoelectric solids are developed by generalizing the Hashin–Shtrikman variational principles (Du and Guo, 2014; Li and Dunn, 2001). Almqvist et al. (2008) derived lower and upper bounds corresponding to the homogenized variational problem including the variational formulation associated with the unstationary Reynolds equation. Quang and He (2008) obtained bounds for composites with coherent imperfect interfaces. Tang and Yu (2008) used asymptotic model for heterogeneous composites. Benveniste and Milton (2010a,2010b) showed the relation of the generalized self-consistent and the Hashin–Shtrikman bounds. Brisard et al. (2010) used the recently developed variational framework for polarization methods in nanocomposites to determine lower bound on the shear modulus of a nanocomposite with monosized, spherical inclusions. Variational bounds for the effective electroelastic moduli of multiphase piezoelectric composites with thin piezoelectric interphase are derived by Wan et al. (2012). Idiart (2013) showed that the bounds are still attained by sequentially laminated constructions when one of the phases is non-linear. An asymptotic micromechanics modeling is developed by Yifeng et al. (2013) for heterogeneous magnetostrictive composite materials. Rigorous bounds for negative stiffness phases are obtained by Kochmann and Milton (2014). Shi et al. (2014) established variational bounds for the effective electroelastic moduli of piezoelectric composites with electromechanical coupling spring-type interfaces. Böhlke and Lobos (2014)gaveHashin–Shtrikman bounds of aggregates of cubic crystals that are explicitly represented in terms of tensorial texture coefficients for arbitrary crystallographic textures and isotropic two-point statistics. Gu and Qin (2014) established the bounds for the effective properties of an inhomogeneous material with coherent imperfect interfaces in the setting of piezoelectricity.

In this work, analytical and semi-analytical upper and lower bounds for effective moduli of transversely isotropic piezoelectric materials are carried out by generalizing the Hashin–Shtrikman variational principle to the coupled problems of piezoelectricity. Incremental downstream (DS) and upstream (US) bounds are introduced herein. The DS and the US are both analyzed here to express two approaches of incremental bounds. Moreover, these incremental bounds are more suited for material design, as we can obtain tighter bounds. The gap between lower and upper bounds is sufficient to characterize with good precision and the overall behavior of heterogeneous material. Then, it is possible to construct materials in the exact solution reaching these bounds. Based on some block matrix formulations, analytical and semi-analytical closed-form expressions are obtained for coupled and uncoupled piezoelectric coefficients. The computation of bounds for composites with ellipsoidal inhomogeneities is presented with a new expression of the electroelastic Eshelby tensor for ellipsoidal inclusions that takes advantage of existing expressions of the Eshelby tensor in the literature. Numerical results are performed and comparisons between predictions obtained by the developed models and the self-consistent method are presented for transversely isotropic piezoelectric materials with various types and shapes of inclusions. The used self-consistent method is the effective medium method (EMM).

Problem formulation

The electromechanical behavior of piezoelectric composite materials is a coupled phenomenon leading to a multifunctional material coupling stress, strain, electric field, and electric displacement. The constitutive piezoelectric equation relating the strain and electric displacement $(ε, D)$ to the stress and electric field $(σ, E)$ is used and expressed by

\begin{matrix} ε_{ij} = S_{ijkl} \cdot σ_{kl} + d_{ijk} \cdot E_{k} \\ D_{i} = d_{ikl} \cdot σ_{kl} + τ_{ik} \cdot E_{k} \end{matrix}

(1)

where S_ijkl, d_ijk, and τ_ik are the elastic compliance tensor (measured in a constant electric field), the piezoelectric tensor, and the dielectric tensor (measured at a constant stress), respectively.

For convenience, the following condensed generalized field notations are adopted

X_{iJ} = P_{iJKl} Y_{Kl}, or Y_{Kl} = P_{iJKl}^{- 1} X_{iJ}

(2)

Where

\begin{matrix} X_{iJ} = {\begin{matrix} ε_{ij} & J = 1, 2, 3 \\ D_{i} & J = 4 \end{matrix} \\ Y_{Ji} = {\begin{matrix} σ_{ji} & J = 1, 2, 3 \\ E_{i} & J = 4 \end{matrix} \end{matrix}

(3)

P_{iJKl} = {\begin{matrix} S_{ijkl} \\ d_{ijl} \\ d_{ikl} \\ τ_{il} \end{matrix} \begin{matrix} J, K = 1, 2, 3 \\ J = 1, 2, 3; K = 4 \\ J = 4; K = 1, 2, 3 \\ J = 4; K = 4 \end{matrix}

(4)

For the heterogeneous materials considered here, the effective electroelastic constitutive equation is defined in a statistical sense, under the assumption of macroscopic homogeneity

Y_{Kl}^{0} = 〈 P_{iJKl}^{- 1} 〉 X_{iJ}^{0}; X_{iJ}^{0} = {〈 P_{iJKl}^{- 1} 〉}^{- 1} Y_{Kl}^{0}

(5)

The expression relating the electroelastic moduli $P_{iJKl}$ and $E_{iJKl}$ , used by Dunn and Taya (1993b), can be easily obtained. Based on the matrix notation, the constitutive equations of piezoelectricity can be expressed by

\begin{matrix} [\begin{matrix} σ_{ji} \\ D_{i} \end{matrix}] = [\begin{matrix} C^{E} & e \\ e^{t} & - κ^{ε} \end{matrix}] [\begin{matrix} ε_{kl} \\ - E_{k} \end{matrix}] \\ [\begin{matrix} ε_{ij} \\ D_{i} \end{matrix}] = P [\begin{matrix} σ_{kl} \\ E_{k} \end{matrix}], where P = [\begin{matrix} S^{E} & d \\ d^{t} & τ^{σ} \end{matrix}] \end{matrix}

(6)

For the sake of clarity, we will omit the superscripts.

The following classical inversion relationships are thus resulted

[\begin{matrix} C & e \\ e^{t} & - κ \end{matrix}] = [\begin{matrix} S^{- 1} & S^{- 1} d \\ d^{t} S^{- 1} & - τ + d^{t} S^{- 1} d \end{matrix}]

(7)

P = [\begin{matrix} S & d \\ d^{t} & τ \end{matrix}] = [\begin{matrix} C^{- 1} & C^{- 1} e \\ e^{t} C^{- 1} & κ + e^{t} C^{- 1} e \end{matrix}]

(8)

The strain and electric field are assumed derivable from the elastic displacement vector u and electric potential φ by the following compatibility equations

ε_{mn} = \frac{1}{2} (u_{m, n} + u_{n, m})

(9a)

E_{n} = - ϕ_{n}

(9b)

The stress and electric displacement fields should verify the equilibrium equations, which in the absence of body forces and free charge are given by the following partial differential system

σ_{ij, j} = 0

(10a)

D_{i, i} = 0

(10b)

Mathematical bounds modeling

Different micromechanical models are used to predict effective moduli of heterogeneous piezoelectric materials. Although these models give direct estimates of effective moduli in terms of composite microstructures, they do not give similar predictions in the most cases and particularly for large volume fractions of inclusions or large contrast of heterogeneities (see for instance Bakkali et al., 2011, 2013; Dunn and Taya, 1993b; El Ouafi et al., 2015; Fakri et al., 2003). It is well known that variational bounds provide good estimations for effective moduli of heterogeneous materials and present a rigorous approach for micromechanical approximations. For this reason, this article is mainly focused on these bounds and their classical formulations are briefly recalled here.

Let V denote the volume of a heterogeneous solid bounded by a surface denoted by δV. We first consider the uniform boundary condition defined by

u_{j} = u_{j}^{0}, x \in δ V_{m 1}; σ_{ij} = σ_{ij}^{0}, x \in δ V_{m 2}

(11a)

D_{j} = D_{j}^{0}, x \in δ V_{e 1}; ϕ = ϕ^{0}, x \in δ V_{e 2}

(11b)

where u_j and $ϕ$ are the elastic displacement and electric potential, respectively, and δV_m₁, δV_m₂, δV_e₁, and δV_e₂ are the sub-regions of δV that satisfy the following requirements

\begin{matrix} δ V_{m 1} \cup δ V_{m 2} = δ V; δ V_{m 1} \cap δ V_{m 2} = \emptyset \\ δ V_{e 1} \cup δ V_{e 2} = δ V; δ V_{e 1} \cap δ V_{e 2} = \emptyset \end{matrix}

(12)

If one considers a piezoelectric field Y_kl, defined in equation (2), that satisfies the specified boundary conditions, the elastic and electrostatic equilibrium equations on Y_Kl(x), the potential energy is defined by Li and Dunn (2001)

\begin{matrix} Φ ({\tilde{Y}}_{Ji}) = & \frac{1}{2} \int_{V} {\tilde{Y}}_{Ji} P_{iJKl} {\tilde{Y}}_{Kl} dV - \int_{δ V_{m 1}} ({\tilde{σ}}_{ij} n_{j}) u_{i}^{0} dSa \\ + \int_{δ V_{m 1}} (D_{i}^{0} n_{j}) \tilde{ϕ} dSa \end{matrix}

(13)

where n_j is the normal to the surface and ${\tilde{Y}}_{Ji}$ , ${\tilde{σ}}_{ij}$ , and $\tilde{ϕ}$ are, respectively, an admissible field, a trial elastic stress, and an admissible electric potential that satisfy the specified boundary conditions, equilibrium equations, and gradient equations on $Y_{Kl} (x)$ . The potential $Φ ({\tilde{Y}}_{Ji})$ is minimized by the actual field, on a condition that electroelastic moduli $P_{iJKl}$ are positive definite. A proof of this theorem is given by Li and Dunn (2001).

Let the applied traction and electric potential be prescribed on the boundary S of the heterogeneous solid

Y_{Ji} (x) = Y_{Ji}^{0}, x \in δ V

(14)

where $Y_{Ji}^{0}$ is a uniform stress and electric field. These boundary conditions correspond to

\begin{matrix} δ V_{m 2} = δ V, δ V_{m 1} = \emptyset \\ δ V_{e 2} = δ V, δ V_{e 1} = \emptyset \end{matrix}

(15)

Based on the average stress and electric field theorem for heterogeneous electroelastic solids (Dunn and Taya, 1993b), one gets $〈 Y_{Ji} 〉 = \frac{1}{V} \int_{V} Y_{Ji} dV = Y_{Ji}^{0}$ . Here, $〈〉 = (1 / V) \int_{V} () dV$ refers to the average of a quantity, over the volume of the heterogeneous materials. The resulting internal energy density in the heterogeneous solid can be written as

U = \frac{1}{2} 〈 Y_{Ji} P_{iJKl} Y_{Kl} 〉 = \frac{1}{2} 〈 Y_{Ji} 〉 {\bar{P}}_{iJKl} 〈 Y_{Kl} 〉

(16)

where ${\bar{P}}_{iJKl}$ is the effective electroelastic moduli of the heterogeneous material.

According to the minimum potential energy theorem, the following inequality holds for any fields that satisfy the boundary conditions

2 U \leq 〈 {\tilde{Y}}_{Ji} P_{iJKl} {\tilde{Y}}_{Kl} 〉

(17)

Substituting equation (16) into equation (17) yields

〈 Y_{Ji} 〉 {\bar{P}}_{iJKl} 〈 Y_{Kl} 〉 \leq 〈 {\tilde{Y}}_{Ji} P_{iJKl} {\tilde{Y}}_{Kl} 〉

(18)

By the Legendre transformation, one can also derive (Li and Dunn, 2001)

〈 Y_{Ji} 〉 {\bar{P}}_{iJKl} 〈 Y_{Kl} 〉 \geq 〈 {\tilde{X}}_{iJ} (2 Y_{Ji}^{0} - P_{iJKl}^{- 1} {\tilde{X}}_{Kl}) 〉

(19)

Equations (18) and (19) give variational bounds for the effective electroelastic moduli by choosing suitable fields that satisfy the boundary conditions

〈 X_{iJ} (2 Y_{Ji}^{0} - P_{iJKl}^{- 1} X_{Kl}) 〉 \leq 〈 Y_{Ji} 〉 {\bar{P}}_{iJKl} 〈 Y_{Kl} 〉 \leq 〈 Y_{Ji} P_{iJKl} Y_{Kl} 〉

(20)

The main objectives of this work are the development of analytical rigorous bounds based on the Hashin–Shtrikman approach as well as on an incremental procedure. Closed-form analytical and semi-analytical formulations are elaborated based on elegant block matrix representations. A brief recall of the classical bounds is first presented.

Voigt–Reuss bounds

The Voigt–Reuss bounds are known to provide an accurate estimate of effective elastic moduli of heterogeneous solids when the electroelastic discrepancy between the different phases in a multiphase composite is very small. Equation (20) shows that by choosing appropriate trial fields, one can give the bounds for the effective electroelastic modulus ${\bar{P}}_{iJKl}$ . By considering the traction–electric potential boundary conditions and elastic displacement–electric displacement boundary conditions in a parallel manner, the uniform fields ${\tilde{Y}}_{Ji} = Y_{Ji}^{0}$ and ${\tilde{X}}_{Ji} = X_{Ji}^{0}$ would be compatible with the boundary conditions. Variational bounds can be thus simplified by choosing the uniform field compatible with the boundary conditions.

Substituting equations $〈 Y_{Ji} 〉 = Y_{Ji}^{0}, 〈 X_{Ji} 〉 = X_{Ji}^{0}$ into equation (20) yields

2 X_{iJ}^{0} Y_{Ji}^{0} - X_{iJ}^{0} 〈 P_{iJKl}^{- 1} 〉 X_{Kl}^{0} \leq Y_{Ji}^{0} {\bar{P}}_{iJKl} Y_{Kl}^{0} \leq Y_{Ji}^{0} 〈 P_{iJKl} 〉 Y_{Kl}^{0}

(21)

Substituting equation (5) into equation (21) yields

\begin{matrix} 2 Y_{Kl}^{0} {〈 P_{iJKl}^{- 1} 〉}^{- 1} Y_{Ji}^{0} - Y_{Kl}^{0} {〈 P_{iJKl}^{- 1} 〉}^{- 1} Y_{Ji}^{0} \\ \leq Y_{Ji}^{0} {\bar{P}}_{iJKl} Y_{Kl}^{0} \leq Y_{Ji}^{0} 〈 P_{iJKl} 〉 Y_{Kl}^{0} \end{matrix}

(22)

This leads to the following bounds of ${\bar{P}}_{iJKl}$

{〈 P_{iJKl}^{- 1} 〉}^{- 1} \leq {\bar{P}}_{iJKl} \leq 〈 P_{iJKl} 〉

(23)

where ${〈 P_{iJKl}^{- 1} 〉}^{- 1}$ is called the Reuss lower bound and $〈 P_{iJKl} 〉$ is called the Voigt upper bound. These bounds are called Voigt–Reuss effective bounds.

If the electroelastic modulus of the rth phase of a specified volume V_r is uniform and given by $P_{iJKl}^{r}$ , where r = 1, …, N (r = 0 corresponds to the matrix), then the Voigt upper bound can be simplified to

〈 P_{iJKl} 〉 = \sum_{r = 1}^{N} c_{r} P_{iJKl}^{r}

(24)

where c_r is the volume fraction of the inhomogeneities with modulus $P_{iJKl}^{r}$ . Equation (24) gives the Voigt upper bound for the effective stiffness tensor of a composite with N distinct phases. Similarly, the Reuss lower bound can be written as

{〈 P_{iJKl}^{- 1} 〉}^{- 1} = {[\sum_{r = 1}^{N} c_{r} (P_{iJKl}^{r})^{- 1}]}^{- 1}

(25)

where $(P_{iJKl}^{r})^{- 1}$ is the compliance modulus for the rth phase.

It should be noted that these bounds are derived in terms of the volume faction of each phase only. Therefore, they are independent on the geometry of the phases and their distribution.

Hashin–Shtrikman bounds

Although the Voigt–Reuss bounds give acceptable bounds for effective electroelastic moduli in some cases, the case of large discrepancy, as is usually the case with piezoelectric composites, the Voigt–Reuss bounds are too wide for practical applications. In this section, rigorous upper and lower bounds for the effective moduli are obtained for heterogeneous piezoelectric materials, by generalizing the Hashin–Shtrikman variational principle to the coupled problems of piezoelectricity.

The used variational bounds were developed by Li and Dunn (2001) and are applicable to statistically homogeneous multiphase composites with any microgeometry and anisotropy, such as composites with misaligned or differently shaped reinforcements, and do not require a particular statistical correlation function. The key to develop the bounds is to find an appropriate field that respects the boundary condition. To do so, we assume that the field tensor is a uniform field on each inhomogeneity including the matrix.

Using the matrix formulation, the tensor $P_{iJKl}^{r}$ is expressed in the matrix form by $P_{r}$ . The expressions of Hashin–Shtrikman bounds are simplified for composites with aligned reinforcements of identical shapes, where the Eshelby tensors for all reinforcements are identical and can be represented as follows (Li and Dunn, 2001)

\begin{matrix} P = & (\sum_{r = 1}^{N} {c_{r} P_{r} [I + H (P_{h})^{- 1} (P_{r} - P_{h})]}^{- 1}) \\ \times {(\sum_{r = 1}^{N} {c_{r} [I + H (P_{h})^{- 1} (P_{r} - P_{h})]}^{- 1})}^{- 1} \end{matrix}

(26)

where $P_{r}$ and $P_{h}$ , are, respectively, the electroelastic moduli of the inclusion related to phase r (r = 2, …, N) and the electroelastic moduli of the reference material. $(P_{r} - P_{h})$ is assumed to be a regular matrix. H is the electroelastic Eshelby tensor, which is used in the computation of the bounds. By choosing reference materials P_h harder or softer (matrix inclusion), one can obtain upper P_u and lower P_l bounds for the electroelastic moduli P.

The expression of H in terms of the Eshelby tensor H^M, given by Dunn and Taya (1993b), is calculated by Li and Dunn (2001)

\begin{matrix} H = & ([\begin{matrix} C & e \\ 0 & - I_{3} \end{matrix}] H^{M} - [\begin{matrix} C & e \\ 0 & 0 \end{matrix}]) \\ \times [\begin{matrix} S (e τ^{- 1} d^{t} - I_{6}) & 0 \\ - τ^{- 1} d^{t} & - I_{3} \end{matrix}] \end{matrix}

(27)

where $I_{6}$ and $I_{3}$ are the identity matrices of orders 6 and 3, respectively.

Based on the fact that

\begin{matrix} [\begin{matrix} C & e \\ 0 & 0 \end{matrix}] [\begin{matrix} S (e τ^{- 1} d^{t} - I_{6}) & 0 \\ - τ^{- 1} d^{t} & - I_{3} \end{matrix}] = - [\begin{matrix} I_{6} & e \\ 0 & 0 \end{matrix}] \\ [\begin{matrix} C & e \\ e^{t} & - κ \end{matrix}] [\begin{matrix} S (e τ^{- 1} d^{t} - I_{6}) & 0 \\ - τ^{- 1} d^{t} & - I_{3} \end{matrix}] = - [\begin{matrix} I_{6} & e \\ 0 & - κ \end{matrix}] \end{matrix}

(28)

The following new expression of Eshelby tensor H is derived

\begin{matrix} H = & - [\begin{matrix} C & e \\ 0 & - I_{3} \end{matrix}] H^{M} {[\begin{matrix} C & e \\ e^{t} & - κ \end{matrix}]}^{- 1} [\begin{matrix} I_{6} & e \\ 0 & - κ \end{matrix}] \\ + [\begin{matrix} I_{6} & e \\ 0 & 0 \end{matrix}] \end{matrix}

(29)

Thus

\begin{matrix} H {[\begin{matrix} S & d \\ d^{t} & τ \end{matrix}]}^{- 1} = - [\begin{matrix} C & e \\ 0 & - I_{3} \end{matrix}] H^{M} {[\begin{matrix} C & e \\ e^{t} & - κ \end{matrix}]}^{- 1} [\begin{matrix} C & 0 \\ e^{t} & - I_{3} \end{matrix}] \\ + [\begin{matrix} C & 0 \\ 0 & 0 \end{matrix}] \end{matrix}

(30)

For a two-phase composite with aligned reinforcements of identical shapes, Hashin–Shtrikman bounds with reference material P_h, given by equation (26), can thus be rewritten as

P = (c_{m} P_{m} A^{- 1} + c_{i} P_{i} B^{- 1}) \times (c_{m} A^{- 1} + c_{i} B^{- 1})^{- 1}

(31)

where (c_m, P_m) and (c_i, P_i) are, respectively, the electroelastic moduli and volume fractions of matrix and inhomogeneities and

A = I + S^{Y} (P_{h})^{- 1} (P_{m} - P_{h})

(32a)

B = I + S^{Y} (P_{h})^{- 1} (P_{i} - P_{h})

(32b)

Equation (31) can thus be re-expressed as

\begin{matrix} P = [P_{m} (c_{m} A^{- 1} + c_{i} B^{- 1}) + c_{i} (P_{i} - P_{m}) B^{- 1}] \\ \times (c_{m} A^{- 1} + c_{i} B^{- 1})^{- 1} \\ P = P_{m} + c_{i} (P_{i} - P_{m}) (c_{m} A^{- 1} B + c_{i} I)^{- 1} \\ P = P_{m} + c_{i} (P_{i} - P_{m}) [c_{m} A^{- 1} (B - A) + I]^{- 1} \end{matrix}

(33)

Based on these formulations, the Hashin–Shtrikman bounds are given by the following expression

\begin{matrix} P = & P_{m} + c_{i} {c_{m} [(S^{Y} P_{h}^{- 1})^{- 1} + (P_{m} - P_{h} {)]}^{- 1} \\ + (P_{i} - P_{m})^{- 1}}^{- 1} \end{matrix}

(34)

As equation (31) is symmetric, with respect to c_i and c_m, one can write

\begin{matrix} P = (c_{m} (P_{m} - P_{i}) A^{- 1} + P_{i} (c_{m} A^{- 1} + c_{i} B^{- 1})) \\ \times (c_{m} A^{- 1} + c_{i} B^{- 1})^{- 1} \\ P = c_{m} (P_{m} - P_{i}) (c_{m} + c_{i} B^{- 1} A)^{- 1} + P_{i} \\ P = P_{i} + c_{m} (P_{m} - P_{i}) (I + c_{i} B^{- 1} (A - B))^{- 1} \end{matrix}

(35)

Thus, this tensor bounds are finally given by the following new formulation

\begin{matrix} P = & P_{i} + c_{m} {c_{i} [(S^{Y} P_{h}^{- 1})^{- 1} + (P_{i} - P_{h} {)]}^{- 1} \\ + (P_{m} - P_{i})^{- 1}}^{- 1} \end{matrix}

(36)

For a two-phase composite, it is generally assumed that the matrix (Epoxy) is softer and inclusion (piezoceramics) is stiffer. Using equation (34), the Hashin–Shtrikman lower bound is obtained by choosing P_h = P_m and given by

P_{l} = P_{m} + f {(P_{i} - P_{m})^{- 1} + (1 - f) S^{Y} P_{m}^{- 1}}^{- 1}

(37)

where f = c_i is the volume fraction of inclusions.

Similarly, using equation (36), the upper bound P_u is expressed by choosing $P_{h} = P_{i}$

P_{u} = P_{i} + (1 - f) {(P_{m} - P_{i})^{- 1} + {fS}^{Y} P_{i}^{- 1}}^{1}

(38)

The last equations lead to the following bound formulations

P_{m}^{- 1} P_{l} = I + f {(P_{i} - P_{m})^{- 1} P_{m} + (1 - f) S^{Y}}^{- 1}

(39)

P_{i}^{- 1} P_{u} = I + (1 - f) {(P_{m} - P_{i})^{- 1} P_{i} + {fS}^{Y}}^{- 1}

(40)

The developed formulations (39) and (40) can be used to obtain numerically the upper and lower bounds.

Note that the choice of the reference material is important for effective bounds. Li and Dunn (2001) stated that as the piezoelectric ceramics are elastically stiffer, but electrically softer than the polymer matrix, the epoxy or the ceramic can be chosen as a comparison material. A numerical procedure has been given by Li and Dunn (2001) for a comparison material. The elastic coefficients of the ceramic or polymer can be kept constant and the dielectric coefficients can be increased or decreased slightly to guarantee that it is the most or least positive definite of the constituent phases.

Mathematical construction of reference material

Because of its importance in the numerical calculations, it is worthwhile to discuss the choice of the reference material. Since the piezoelectric ceramics are elastically stiffer, but electrically softer than the polymer matrix, we cannot directly choose the epoxy or the ceramic as a comparison material. Li and Dunn (2001), however, keep the elastic constants of the ceramic or polymer and increase or decrease their dielectric constants slightly to guarantee that it is the most or least positive definite of the constituent phases. The positive definiteness is verified by computing the eigenvalues. With the comparison material so chosen, the calculation of the bounds can be carried out.

To generate a correct comparison material, a simple mathematical method is given here. Let us define the matrix difference by M = P_i−P_m, where P_m and P_i are, respectively, the electroelastic moduli of matrix and inhomogeneities, and M is assumed to be diagonalizable and its decomposition is given by

M = {LDL}^{t}

(41)

in which L is orthogonal and D is diagonal containing the eigenvalues of M defined by

D = (\begin{matrix} λ_{1} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & λ_{n} \end{matrix})

(42)

By choosing α_i and β_i positive values such that β_i−α_i = λ_i for i = 1, …, n, the matrices

M_{β} = L (\begin{matrix} β_{1} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & β_{n} \end{matrix}) L^{t}

(43)

and

M_{α} = L (\begin{matrix} α_{1} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & α_{n} \end{matrix}) L^{t}

(44)

are positive symmetric such that M_β−M_α = M = P_i−P_m. Then

- M_{α} + P_{m} = - M_{β} + P_{i} = P_{r}

(45)

Finally, the matrices

\begin{matrix} P_{m} - P_{r} = M_{α} \\ P_{i} - P_{r} = M_{β} \end{matrix}

(46)

are symmetric and positive definite.

For the numerical calculation then performed, here, we have chosen α_j = −min(λ_j,0) and β_j = max(λ_j,0), α_j and β_j are then positive values such that β_j−α_j = λ_j for j = 1, …, n. This algorithm leads to mathematical construction of reference material upon the algorithm used by Li and Dunn (2001) to compute eigenvalues of reference material.

Incremental DS and US bounds

To improve the Hashin–Shtrikman bounds, new effective incremental bounds are introduced. The incremental methodological approach consists of replacing a finite increment of the volume fraction of matrix or inclusions in a certain effective medium. This procedure will lead to the so called here incremental bounds.

For a two-phase composite, the obtained composite must be characterized by the concentration f of inclusions. $Δ f = f / s$ is considered as the partial concentration of the inclusions phase and s is the number of steps. The concept of the volume preservation must be used for computing the finite increment of the total volume fraction of reinforcements which will be added at the nth step. This procedure has been followed by Fakri and Azrar (2010) and allows elaborating an incremental self-consistent model for piezocomposites.

Incremental DS bounds

Let us consider that $V_{m}$ and $V_{i}$ are the volumes of matrix and inclusion, respectively, the volume fraction of inclusion is then $f = V_{i} / (V_{i} + V_{m})$ . At the nth step, the volumes of both inclusion and the new resulting composite at the (n−1)th step are, respectively, $Δ V_{i} = Δ f . V_{i}$ and $V_{h}^{n - 1} = V_{m} + (n - 1) Δ V_{i}$ . The expression of the finite volume fraction of inclusions to be injected in the matrix at the nth step is then given by

Δ f_{n} = \frac{Δ V_{i}}{Δ V_{i} + V_{h}^{n - 1}} = \frac{Δ V_{i}}{Δ V_{i} + V_{m} + (n - 1) Δ V_{i}}

(47)

This can be expressed as

\begin{matrix} Δ f_{n} = \frac{V_{i} / s}{V_{i} / s + V_{m} + (n - 1) V_{i} / s} \\ = \frac{V_{i}}{{nV}_{i} + {sV}_{m}} \\ = \frac{V_{i}}{nV + (s - n) V_{m}} \\ = \frac{f}{n + (s - n) (1 - f)} \\ = \frac{Δ f}{1 - (s - n) Δ f} \end{matrix}

(48)

For each increment, the computed bound is taken as the new comparison material and applied to calculate new incremental bounds of the composite materials using equations (39) and (40).

The bounds for the effective properties of the composites, given by the presented incremental process, depend on the number of steps. The corresponding nth step lower bound is thus given by the following relationship

\begin{matrix} P_{dl}^{n} = & P_{dl}^{n - 1} + Δ f_{n} {(P_{i} - P_{dl}^{n - 1})^{- 1} \\ + (1 - Δ f_{n}) S^{Y} (P_{dl}^{n - 1})^{- 1}}^{- 1}, n \geq 1 \end{matrix}

(49)

with $P_{dl}^{0} = P_{m}$ .

Similarly, the nth step upper bound is obtained by interchanging (P_l, Δf_n) and (P_i, 1−Δf_n) and thus expressed by

\begin{matrix} P_{du}^{n} = & P_{i} + (1 - Δ f_{n}) {(P_{du}^{n - 1} - P_{i})^{- 1} \\ + Δ f_{n} S^{Y} (P_{i})^{- 1}}^{- 1}, n \geq 1 \end{matrix}

(50)

with $P_{du}^{0} = P_{m}$ , in which the finite volume fraction to be injected for the matrix in each step is given by equation (48).

Incremental US bounds

Similarly, the expression of US bounds is established by considering at the (n−1)th step the following volume fractions $Δ V_{m} = Δ f \cdot V_{m}$ and $V_{h}^{n - 1} = V_{i} + (n - 1) Δ V_{m}$ . At the nth step, the expression of the finite volume fraction of inclusions to be injected in the matrix is

\begin{matrix} Δ f_{n} = \frac{Δ V_{m}}{Δ V_{m} + V_{h}^{n - 1}} \\ = \frac{Δ V_{m}}{Δ V_{m} + V_{i} + (n - 1) Δ V_{m}} \\ = \frac{Δ f}{1 - (s - n) Δ f} \end{matrix}

(51)

in which $Δ f = \frac{1 - f}{s}$

The resulting US incremental bounds are then expressed as

\begin{matrix} P_{ul}^{n} = & P_{m} + (1 - Δ f_{n}) {(P_{ul}^{n - 1} - P_{m})^{- 1} \\ + Δ f_{n} S^{Y} (P_{m})^{- 1}}^{- 1}, n \geq 1 \end{matrix}

(52a)

\begin{matrix} P_{uu}^{n} = & P_{uu}^{n - 1} + Δ f_{n} {(P_{m} - P_{uu}^{n - 1})^{- 1} \\ + (1 - Δ f_{n}) S^{Y} (P_{uu}^{n - 1})^{- 1}}^{- 1}, n \geq 1 \end{matrix}

(52b)

in which $P_{ul}^{0} = P_{uu}^{0} = P_{i}$ . These bounds are called here the DS and US incremental bounds since the inclusion is the finite volume fraction to be injected in the matrix.

Limit cases

For incremental DS bounds, if the total number of steps s = 1, then Δf_n = Δf = f from equation (48). Equations (49) and (50) are then reduced to the following expressions

\begin{matrix} P_{dl}^{1} = P_{dl}^{0} + Δ f {(P_{i} - P_{dl}^{0})^{- 1} + (1 - Δ f) S^{Y} (P_{dl}^{0})^{- 1}}^{- 1} \\ P_{du}^{1} = P_{i} + (1 - Δ f) {(P_{du}^{0} - P_{i})^{- 1} + Δ {fS}^{Y} (P_{i})^{- 1}}^{- 1} \end{matrix}

(53)

\begin{matrix} P_{dl}^{1} = P_{m} + f {(P_{i} - P_{m})^{- 1} + (1 - f) S^{Y} (P_{m})^{- 1}}^{- 1} \\ P_{du}^{1} = P_{i} + (1 - f) {(P_{m} - P_{i})^{- 1} + {fS}^{Y} (P_{i})^{- 1}}^{- 1} \end{matrix}

(54)

which are the same equations than equation (37) corresponding to the Hashin–Shtrikman model.

Similarly, for US bounds, if the number of steps s = 1, Δf_n = Δf = 1−f, then equations (52a) and (52b) are reduced to the following expressions

\begin{matrix} P_{ul}^{1} = P_{m} + (1 - Δ f) {(P_{ul}^{0} - P_{m}) + Δ {fS}^{Y} (P_{m})^{- 1}}^{- 1} \\ P_{uu}^{1} = P_{uu}^{0} + Δ f {(P_{m} - P_{uu}^{0})^{- 1} + (1 - Δ f) S^{Y} (P_{uu}^{0})^{- 1}}^{- 1} \end{matrix}

(55)

\begin{matrix} P_{ul}^{l} = P_{m} + (1 - Δ f) {(P_{i} - P_{m}) + Δ {fS}^{Y} (P_{m})^{- 1}}^{- 1} \\ P_{uu}^{l} = P_{i} + Δ f {(P_{m} - P_{i})^{- 1} + (1 - Δ f) S^{Y} (P_{i})^{- 1}}^{- 1} \end{matrix}

(56)

which correspond to the Hashin–Shtrikman bounds given in equation (37).

For large n, it is demonstrated that the two DS formulations $P_{dl}^{n}$ and $P_{du}^{n}$ converge to a limit that is named the DS bound and the two US formulations $P_{ul}^{n}$ and $P_{uu}^{n}$ converge to a limit called the US limit bound. These two bound limits are considered here as new DS and US bounds. Figures 1 and 2 present the results obtained for d₃₁ for n = 1 to 100 as well as the convergence with respect to n. It is clearly demonstrated that n = 50 is enough for convergence of the presented processes.

Figure 1.

Effect of the numbers of steps s on the effective piezoelectric modulus d₃₁ based on the incremental downstream upper bounds for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

Figure 2.

Effect of the numbers of steps s on the effective piezoelectric modulus d₃₁ based on the incremental upstream lower bounds for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

The relationships, presented in the previous sub-sections, can be used to predict numerically the effective bounds for the considered heterogeneous materials. Based on the presented relationships, analytical closed-form expressions for some decoupled coefficients will be elaborated in the next section. It should be noted that the choice of matrix and inhomogeneity as reference materials gives simplified relationships.

Analytical and semi-analytical models

One of the main objectives of this article is to elaborate explicit analytical and semi-analytical expression models of bounds for some effective piezoelectric coefficients of the composite as well as to give simplified mathematical formulations for the others. For this aim, simple matrix formulations are developed as well as closed-form relations of bounds for coupled and decoupled coefficients. Transversely isotropic piezoelectric matrix and ellipsoidal transversely isotropic piezo inclusions are considered.

Let us assume that the inclusions and the matrix are transversely isotropic piezoelectric. For the sake of clarity, the used block matrix forms are explicitly given here for transversely isotropic materials, using the 9 × 9 matrix notation

P = [\begin{matrix} S_{11} & S_{12} & S_{13} & 0 & 0 & 0 & 0 & 0 & d_{31} \\ S_{12} & S_{22} & S_{23} & 0 & 0 & 0 & 0 & 0 & d_{32} \\ S_{13} & S_{23} & S_{33} & 0 & 0 & 0 & 0 & 0 & d_{33} \\ 0 & 0 & 0 & S_{44} & 0 & 0 & 0 & d_{15} & 0 \\ 0 & 0 & 0 & 0 & S_{55} & 0 & d_{15} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{66} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{15} & 0 & τ_{11} & 0 & 0 \\ 0 & 0 & 0 & d_{15} & 0 & 0 & 0 & τ_{22} & 0 \\ d_{31} & d_{32} & d_{33} & 0 & 0 & 0 & 0 & 0 & τ_{33} \end{matrix}]

(57)

In this case, the upper and lower electroelastic matrices P_l and P_u have also similar forms

\begin{matrix} P_{k} = [\begin{matrix} S_{11}^{k} & S_{12}^{k} & S_{13}^{k} & 0 & 0 & 0 & 0 & 0 & d_{31}^{k} \\ S_{12}^{k} & S_{22}^{k} & S_{23}^{k} & 0 & 0 & 0 & 0 & 0 & d_{32}^{k} \\ S_{13}^{k} & S_{23}^{k} & S_{33}^{k} & 0 & 0 & 0 & 0 & 0 & d_{33}^{k} \\ 0 & 0 & 0 & S_{44}^{k} & 0 & 0 & 0 & d_{15}^{k} & 0 \\ 0 & 0 & 0 & 0 & S_{55}^{k} & 0 & d_{15}^{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{66}^{k} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{15}^{k} & 0 & τ_{11}^{k} & 0 & 0 \\ 0 & 0 & 0 & d_{15}^{k} & 0 & 0 & 0 & τ_{22}^{k} & 0 \\ d_{31}^{k} & d_{32}^{k} & d_{33}^{k} & 0 & 0 & 0 & 0 & 0 & τ_{33}^{k} \end{matrix}], \\ k = l or u \end{matrix}

(58)

The Eshelby matrix can be also expressed as

S^{Y} = [\begin{matrix} S_{11}^{Y} & S_{12}^{Y} & S_{13}^{Y} & 0 & 0 & 0 & 0 & 0 & S_{19}^{Y} \\ S_{21}^{Y} & S_{22}^{Y} & S_{23}^{Y} & 0 & 0 & 0 & 0 & 0 & S_{29}^{Y} \\ S_{31}^{Y} & S_{32}^{Y} & S_{33}^{Y} & 0 & 0 & 0 & 0 & 0 & S_{39}^{Y} \\ 0 & 0 & 0 & S_{44}^{Y} & 0 & 0 & 0 & S_{48}^{Y} & 0 \\ 0 & 0 & 0 & 0 & S_{55}^{Y} & 0 & S_{57}^{Y} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{66}^{Y} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{75}^{Y} & 0 & S_{77}^{Y} & 0 & 0 \\ 0 & 0 & 0 & S_{84}^{Y} & 0 & 0 & 0 & S_{88}^{Y} & 0 \\ S_{91}^{Y} & S_{92}^{Y} & S_{93}^{Y} & 0 & 0 & 0 & 0 & 0 & S_{99}^{Y} \end{matrix}]

(59)

For the lower and upper bounds $P_{l}$ and $P_{u}$ , the relationships (39) and (40) can be rewritten as

\begin{matrix} P_{m}^{- 1} P_{l} = I_{d} + f \cdot T_{l}^{- 1} \\ P_{i}^{- 1} P_{u} = I_{d} + (1 - f) \cdot T_{u}^{- 1} \end{matrix}

(60)

where

\begin{matrix} T_{l} = (P_{i} - P_{m})^{- 1} P_{m} + (1 - f) S^{Y} \\ T_{u} = (P_{m} - P_{i})^{- 1} P_{i} + {fS}^{Y} \end{matrix}

(61)

Based on these relations, it is easy to demonstrate that the matrices T_k, k = (l, u), can be expressed as

T_{k} = [\begin{matrix} T_{11}^{k} & T_{12}^{k} & T_{13}^{k} & 0 & 0 & 0 & 0 & 0 & T_{19}^{k} \\ T_{21}^{k} & T_{22}^{k} & T_{23}^{k} & 0 & 0 & 0 & 0 & 0 & T_{29}^{k} \\ T_{31}^{k} & T_{32}^{k} & T_{33}^{k} & 0 & 0 & 0 & 0 & 0 & T_{39}^{k} \\ 0 & 0 & 0 & T_{44}^{k} & 0 & 0 & 0 & T_{48}^{k} & 0 \\ 0 & 0 & 0 & 0 & T_{55}^{k} & 0 & T_{57}^{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & T_{66}^{k} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & T_{75}^{k} & 0 & T_{77}^{k} & 0 & 0 \\ 0 & 0 & 0 & T_{84}^{k} & 0 & 0 & 0 & T_{88}^{k} & 0 \\ T_{91}^{k} & T_{92}^{k} & T_{93}^{k} & 0 & 0 & 0 & 0 & 0 & T_{99}^{k} \end{matrix}]

(62)

Based on the block matrix notations, the matrices associated to P_k, P_h, and T are expressed as

\begin{matrix} P_{k} = [\begin{matrix} A_{k} & 0 & M_{k} \\ 0^{t} & B_{k} & 0 \\ M_{k}^{t} & 0^{t} & C_{k} \end{matrix}] \\ P_{h} = [\begin{matrix} A_{h} & 0 & M_{h} \\ 0^{t} & B_{h} & 0 \\ M_{h}^{t} & 0^{t} & C_{h} \end{matrix}] \\ T_{k} = [\begin{matrix} T_{k}^{33} & 0 & T_{k}^{31} \\ 0^{t} & T_{k}^{55} & 0 \\ T_{k}^{13} & 0^{t} & T_{k}^{11} \end{matrix}] \end{matrix}

(63)

where the subscripts k is given by k = l for lower case and k = u for upper case.

Expressions of the resulting sub-matrices are given below

A_{k} = [\begin{matrix} S_{11}^{k} & S_{12}^{k} & S_{13}^{k} \\ S_{12}^{k} & S_{22}^{k} & S_{23}^{k} \\ S_{13}^{k} & S_{23}^{k} & S_{33}^{k} \end{matrix}]

(64a)

M_{k} = [\begin{matrix} d_{31}^{k} \\ d_{32}^{k} \\ d_{33}^{k} \end{matrix}]

(64b)

M_{k}^{t} = [\begin{matrix} d_{31}^{k} & d_{32}^{k} & d_{33}^{k} \end{matrix}]

(64c)

B_{k} = [\begin{matrix} S_{44}^{k} & 0 & 0 & 0 & d_{24}^{k} \\ 0 & S_{55}^{k} & 0 & d_{15}^{k} & 0 \\ 0 & 0 & S_{66}^{k} & 0 & 0 \\ 0 & d_{15}^{k} & 0 & τ_{11}^{k} & 0 \\ d_{24}^{k} & 0 & 0 & 0 & τ_{22}^{k} \end{matrix}]

(64d)

C_{k} = [τ_{33}^{k}]

(64e)

A_{h} = [\begin{matrix} S_{11}^{h} & S_{12}^{h} & S_{13}^{h} \\ S_{12}^{h} & S_{22}^{h} & S_{23}^{h} \\ S_{13}^{h} & S_{23}^{h} & S_{33}^{h} \end{matrix}]

(65a)

M_{h} = [\begin{matrix} d_{31}^{h} \\ d_{32}^{h} \\ d_{33}^{h} \end{matrix}]

(65b)

M_{h}^{t} = [\begin{matrix} d_{31}^{h} & d_{32}^{h} & d_{33}^{h} \end{matrix}]

(65c)

B_{h} = [\begin{matrix} S_{44}^{h} & 0 & 0 & 0 & d_{15}^{h} \\ 0 & S_{55}^{h} & 0 & d_{15}^{h} & 0 \\ 0 & 0 & S_{66}^{h} & 0 & 0 \\ 0 & d_{15}^{h} & 0 & τ_{11}^{h} & 0 \\ d_{15}^{h} & 0 & 0 & 0 & τ_{22}^{h} \end{matrix}]

(65d)

C_{h} = [τ_{33}^{h}]

(65e)

T_{k}^{33} = [\begin{matrix} T_{11}^{k} & T_{12}^{k} & T_{13}^{k} \\ T_{21}^{k} & T_{22}^{k} & T_{23}^{k} \\ T_{31}^{k} & T_{32}^{k} & T_{33}^{k} \end{matrix}]

(66a)

T_{k}^{31} = [\begin{matrix} T_{19}^{k} \\ T_{29}^{k} \\ T_{39}^{k} \end{matrix}]

(66b)

T_{k}^{13} = [\begin{matrix} T_{91}^{k} & T_{92}^{k} & T_{93}^{k} \end{matrix}]

(66c)

T_{k}^{55} = [\begin{matrix} T_{44}^{k} & 0 & 0 & 0 & T_{48}^{k} \\ 0 & T_{55}^{k} & 0 & T_{57}^{k} & 0 \\ 0 & 0 & T_{66}^{k} & 0 & 0 \\ 0 & T_{75}^{k} & 0 & T_{77}^{k} & 0 \\ T_{84}^{k} & 0 & 0 & 0 & T_{88}^{k} \end{matrix}]

(66d)

T_{k}^{11} = [T_{99}^{k}]

(66e)

The inverse of the previous matrices is formally obtained by the following relation

\begin{matrix} {[\begin{matrix} A & 0 & M \\ 0^{t} & B & 0 \\ M^{t} & 0^{t} & C \end{matrix}]}^{- 1} \\ = [\begin{matrix} (A - M C^{- 1} M^{t})^{- 1} & 0 & - A^{- 1} M (C - M^{t} A^{- 1} M)^{- 1} \\ 0^{t} & B^{- 1} & 0 \\ - C^{- 1} M^{t} (A - M C^{- 1} M^{t})^{- 1} & 0^{t} & (C - M^{t} A^{- 1} M)^{- 1} \end{matrix}] \end{matrix}

(67)

These mathematical developments lead to the following simplified main relationships

B_{m}^{- 1} B_{l} = I_{5} + f (T_{l}^{55})^{- 1}

(68a)

B_{i}^{- 1} B_{u} = I_{5} + (1 - f) (T_{u}^{55})^{- 1}

(68b)

{[\begin{matrix} A_{l} & M_{l} \\ M_{l}^{t} & C_{l} \end{matrix}]}^{- 1} [\begin{matrix} A_{m} & M_{m} \\ M_{m}^{t} & C_{m} \end{matrix}] = [\begin{matrix} I_{3} & 0 \\ 0 & I_{1} \end{matrix}] + f {[\begin{matrix} T_{l}^{33} & T_{l}^{31} \\ T_{l}^{13} & T_{l}^{11} \end{matrix}]}^{- 1}

(69a)

{[\begin{matrix} A_{u} & M_{u} \\ M_{u}^{t} & C_{u} \end{matrix}]}^{- 1} [\begin{matrix} A_{i} & M_{i} \\ M_{i}^{t} & C_{i} \end{matrix}] = [\begin{matrix} I_{3} & 0 \\ 0 & I_{1} \end{matrix}] + f {[\begin{matrix} T_{u}^{33} & T_{u}^{31} \\ T_{u}^{13} & T_{u}^{11} \end{matrix}]}^{- 1}

(69b)

in which $I_{1}$ , $I_{3}$ , and $I_{5}$ are, respectively, the identity matrices of orders 1, 3, and 5. These block matrix formulations are used here to elaborate analytical effective models for decoupled coefficients and simplified formulations for coupled ones.

Based on some mathematical developments and on equations (68a) and (68b), the following matrix form is resulted

\begin{matrix} [\begin{matrix} \frac{(S_{44})_{k} τ_{22}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} - \frac{(d_{24})_{k} d_{15}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} & 0 & 0 & 0 & \frac{(d_{24})_{k} τ_{22}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} - \frac{(τ_{22})_{k} d_{15}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} \\ 0 & \frac{(S_{44})_{k} τ_{22}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} - \frac{(d_{15})_{k} d_{15}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} & 0 & \frac{(d_{15})_{k} τ_{22}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} - \frac{(τ_{11})_{k} d_{15}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} & 0 \\ 0 & 0 & \frac{(S_{66})_{k}}{S_{66}^{h}} & 0 & 0 \\ 0 & - \frac{(S_{44})_{k} d_{15}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} + \frac{(d_{15})_{k} S_{44}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} & 0 & \frac{(d_{15})_{k} d_{15}^{h}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} - \frac{S_{44}^{h} (τ_{11})_{k}}{τ_{11}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} & 0 \\ - \frac{(S_{44})_{k} d_{15}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} + \frac{(d_{24})_{k} S_{44}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} & 0 & 0 & 0 & - \frac{(d_{24})_{k} d_{15}^{h}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} + \frac{S_{44}^{h} (τ_{22})_{k}}{τ_{22}^{h} S_{44}^{h} - (d_{15}^{h})^{2}} \end{matrix}] = \\ [\begin{matrix} 1 + f_{k} \frac{T_{88}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} & 0 & 0 & 0 & - f_{k} \frac{T_{48}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} \\ 0 & 1 + f_{k} \frac{T_{77}^{k}}{T_{77}^{k} T_{55}^{k} - T_{57}^{k} T_{75}^{k}} & 0 & - f_{k} \frac{T_{57}^{k}}{T_{77}^{k} T_{55}^{k} - T_{57}^{k} T_{75}^{k}} & 0 \\ 0 & 0 & 1 + f_{k} \frac{1}{T_{66}^{k}} & 0 & 0 \\ 0 & - f_{k} \frac{T_{75}^{k}}{T_{77}^{k} T_{55}^{k} - T_{57}^{k} T_{75}^{k}} & 0 & 1 + f_{k} \frac{T_{55}^{k}}{T_{77}^{k} T_{55}^{k} - T_{57}^{k} T_{75}^{k}} & 0 \\ - f_{k} \frac{T_{84}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} & 0 & 0 & 0 & 1 + f_{k} \frac{T_{44}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} \end{matrix}] \end{matrix}

(70)

where

f_{k} = {\begin{matrix} f if k = l \\ 1 - f if k = u \end{matrix}, h = {\begin{matrix} m if k = l \\ i if k = u \end{matrix}

(71)

The matrix formulation (70) permits to get analytical effective bound model for all coefficients of the central matrix B_k, that is, S₄₄, S₅₅, S₆₆, τ₁₁, τ₂₂, and d₁₅. The following simplified bound relations are explicitly deduced

\begin{matrix} \frac{(S_{44})_{k} τ_{22}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} - \frac{(d_{24})_{k} d_{15}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} \\ = 1 + f_{k} \frac{T_{88}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} \end{matrix}

(72a)

\begin{matrix} \frac{(d_{24})_{k} τ_{22}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} - \frac{(τ_{22})_{k} d_{15}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} \\ = - f_{k} \frac{T_{48}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} \end{matrix}

(72b)

\begin{matrix} - \frac{(S_{44})_{k} d_{15}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} + \frac{(d_{24})_{k} S_{44}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} \\ = - f_{k} \frac{T_{84}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} \end{matrix}

(72c)

\begin{matrix} - \frac{(d_{24})_{k} d_{15}^{m}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} + \frac{S_{44}^{m} (τ_{22})_{k}}{τ_{22}^{m} S_{44}^{m} - (d_{15}^{m})^{2}} \\ = 1 + f_{k} \frac{T_{44}^{k}}{T_{88}^{k} T_{44}^{k} - T_{48}^{k} T_{84}^{k}} \end{matrix}

(72d)

For closed-form expressions, the required $T_{ij}^{k}$ associated to lower and upper bounds have to be explicitly given. For lower bounds, $T_{ij}^{l}$ are given by the following relationships

\begin{matrix} T_{44}^{l} = \frac{S_{44}^{m} (τ_{22}^{i} - τ_{22}^{m}) - d_{15}^{m} (d_{15}^{i} - d_{15}^{m})}{(τ_{22}^{i} - τ_{22}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + 2 (1 - f) H_{44} \\ T_{88}^{l} = \frac{τ_{22}^{m} (S_{44}^{i} - S_{44}^{m}) - d_{15}^{m} (d_{15}^{i} - d_{15}^{m})}{(τ_{22}^{i} - τ_{22}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + (1 - f) H_{88} \\ T_{48}^{l} = - \frac{d_{15}^{i} τ_{22}^{m} - d_{15}^{m} τ_{22}^{i}}{(τ_{22}^{i} - τ_{22}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + (1 - f) H_{48} \\ T_{84}^{l} = \frac{d_{15}^{m} S_{44}^{i} - d_{15}^{i} S_{44}^{m}}{(τ_{22}^{i} - τ_{22}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + (1 - f) H_{84} \end{matrix}

(73)

For upper bounds, $T_{ij}^{u}$ are expressed as

\begin{matrix} T_{44}^{u} = \frac{S_{44}^{i} (τ_{22}^{m} - τ_{22}^{i}) - d_{15}^{i} (d_{15}^{m} - d_{15}^{i})}{(τ_{22}^{m} - τ_{22}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + 2 {fH}_{44} \\ T_{88}^{u} = \frac{τ_{22}^{i} (S_{44}^{m} - S_{44}^{i}) - d_{15}^{i} (d_{15}^{m} - d_{15}^{i})}{(τ_{22}^{m} - τ_{22}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + {fH}_{88} \\ T_{48}^{u} = - \frac{d_{15}^{m} τ_{22}^{i} - d_{15}^{i} τ_{22}^{m}}{(τ_{22}^{m} - τ_{22}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + {fH}_{48} \\ T_{84}^{u} = \frac{d_{15}^{i} S_{44}^{m} - d_{15}^{m} S_{44}^{i}}{(τ_{22}^{m} - τ_{22}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + {fH}_{84} \end{matrix}

(74)

Analytical models for lower bounds of shear moduli $S_{44}^{l}$ and $d_{15}^{l}$ and dielectric modulus $τ_{22}^{l}$ are then given by

S_{44}^{l} = S_{44}^{m} + f \frac{S_{44}^{m} T_{88}^{l} - d_{15}^{m} T_{84}^{l}}{T_{88}^{l} T_{44}^{l} - T_{48}^{l} T_{84}^{l}}

(75a)

τ_{22}^{l} = τ_{22}^{m} + f \frac{τ_{22}^{m} T_{44}^{l} - d_{15}^{m} K_{48}^{l}}{T_{88}^{l} T_{44}^{l} - T_{48}^{l} T_{84}^{l}}

(75b)

d_{24}^{l} = d_{15}^{m} + f \frac{d_{15}^{m} T_{88}^{l} - τ_{22}^{m} T_{84}^{l}}{T_{88}^{l} T_{44}^{l} - T_{48}^{l} T_{84}^{l}}

(75c)

Analytical models for the shear components $S_{44}^{u}$ and $d_{15}^{u}$ and the dielectric modulus $τ_{22}^{u}$ for upper bounds are

S_{44}^{u} = S_{44}^{i} + (1 - f) \frac{S_{44}^{i} T_{88}^{u} - d_{15}^{i} T_{84}^{u}}{T_{88}^{u} T_{44}^{u} - T_{48}^{u} T_{84}^{u}}

(76a)

τ_{22}^{u} = τ_{22}^{i} + (1 - f) \frac{τ_{22}^{i} T_{44}^{u} - d_{15}^{i} T_{48}^{u}}{T_{88}^{u} T_{44}^{u} - T_{48}^{u} T_{84}^{u}}

(76b)

d_{24}^{u} = d_{15}^{i} + (1 - f) \frac{d_{15}^{i} T_{88}^{u} - τ_{22}^{i} T_{84}^{u}}{T_{88}^{u} T_{44}^{u} - T_{48}^{u} T_{84}^{u}}

(76c)

Similarly, using the previous procedures to obtain the lower and upper bounds of the coefficients $S_{55}$ , $d_{15}$ , and $τ_{11}$ , the following closed-form relationships are obtained

S_{55}^{l} = S_{55}^{m} + f \frac{S_{55}^{m} T_{77}^{l} - d_{15}^{m} T_{75}^{l}}{T_{77}^{l} K_{55}^{l} - T_{57}^{l} T_{75}^{l}}

(77a)

S_{55}^{u} = S_{55}^{i} + (1 - f) \frac{S_{55}^{i} T_{77}^{u} - d_{15}^{i} T_{75}^{u}}{T_{77}^{u} T_{55}^{u} - T_{57}^{u} T_{75}^{u}}

(77b)

τ_{11}^{l} = τ_{11}^{m} + f \frac{τ_{11}^{m} T_{55}^{l} - d_{15}^{m} T_{57}^{l}}{T_{77}^{l} T_{55}^{l} - T_{57}^{l} T_{75}^{l}}

(78a)

τ_{11}^{u} = τ_{11}^{i} + (1 - f) \frac{τ_{11}^{i} T_{55}^{u} - d_{15}^{i} T_{57}^{u}}{T_{77}^{u} T_{55}^{u} - T_{57}^{u} T_{75}^{u}}

(78b)

d_{15}^{l} = d_{15}^{m} + f \frac{d_{15}^{m} T_{77}^{l} - τ_{11}^{m} T_{75}^{l}}{T_{77}^{l} T_{55}^{l} - T_{57}^{l} T_{75}^{l}}

(79a)

d_{15}^{u} = d_{15}^{i} + (1 - f) \frac{d_{15}^{i} T_{77}^{u} - τ_{11}^{i} T_{75}^{u}}{T_{77}^{u} T_{55}^{u} - T_{57}^{u} T_{75}^{u}}

(79b)

where

\begin{matrix} T_{55}^{l} = \frac{S_{44}^{m} (τ_{11}^{i} - τ_{11}^{m}) - d_{15}^{m} (d_{15}^{i} - d_{15}^{m})}{(τ_{11}^{i} - τ_{11}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + 2 (1 - f) H_{55} \\ T_{77}^{l} = \frac{τ_{11}^{m} (S_{44}^{i} - S_{44}^{m}) - d_{15}^{m} (d_{15}^{i} - d_{15}^{m})}{(τ_{11}^{i} - τ_{11}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + (1 - f) H_{77} \\ T_{57}^{l} = - \frac{d_{15}^{i} τ_{11}^{m} - d_{15}^{m} τ_{11}^{i}}{(τ_{11}^{i} - τ_{11}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + (1 - f) H_{57} \\ T_{75}^{l} = \frac{d_{15}^{m} S_{44}^{i} - d_{15}^{i} S_{44}^{m}}{(τ_{11}^{i} - τ_{11}^{m}) (S_{44}^{i} - S_{44}^{m}) - (d_{15}^{i} - d_{15}^{m})^{2}} + (1 - f) H_{75} \end{matrix}

(80)

and

\begin{matrix} T_{55}^{u} = \frac{S_{44}^{i} (τ_{11}^{m} - τ_{11}^{i}) - d_{15}^{i} (d_{15}^{m} - d_{15}^{i})}{(τ_{11}^{m} - τ_{11}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + 2 {fH}_{55} \\ T_{77}^{u} = \frac{τ_{11}^{i} (S_{44}^{m} - S_{44}^{i}) - d_{15}^{i} (d_{15}^{m} - d_{15}^{i})}{(τ_{11}^{m} - τ_{11}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + {fH}_{77} \\ T_{57}^{u} = - \frac{d_{15}^{m} τ_{11}^{i} - d_{15}^{i} τ_{11}^{m}}{(τ_{11}^{m} - τ_{11}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + {fH}_{57} \\ T_{75}^{u} = \frac{d_{15}^{i} S_{44}^{m} - d_{15}^{m} S_{44}^{i}}{(τ_{11}^{m} - τ_{11}^{i}) (S_{44}^{m} - S_{44}^{i}) - (d_{15}^{m} - d_{15}^{i})^{2}} + {fH}_{75} \end{matrix}

(81)

The expressions of lower and upper bounds for the shear modulus $S_{66}$ are

\begin{matrix} \frac{S_{66}^{l}}{S_{66}^{m}} = 1 + f \frac{1}{\frac{S_{66}^{m}}{(S_{66}^{i} - S_{66}^{m})} + 2 (1 - f) H_{66}} \\ \frac{S_{66}^{u}}{S_{66}^{i}} = 1 + (1 - f) \frac{1}{\frac{S_{66}^{i}}{(S_{66}^{m} - S_{66}^{i})} + 2 {fH}_{66}} \end{matrix}

(82)

The prediction of lower and upper bounds of the other effective coefficients can be obtained by the following simplified matrix relationships

{[\begin{matrix} A_{l} & M_{l} \\ M_{l}^{t} & C_{l} \end{matrix}]}^{- 1} [\begin{matrix} A_{m} & M_{m} \\ M_{m}^{t} & C_{m} \end{matrix}] = [\begin{matrix} I_{3} & 0 \\ 0 & I_{1} \end{matrix}] + f {[\begin{matrix} T_{l}^{33} & T_{l}^{31} \\ T_{l}^{13} & T_{l}^{11} \end{matrix}]}^{- 1}

(83a)

\begin{matrix} {[\begin{matrix} A_{u} & M_{u} \\ M_{u}^{t} & C_{u} \end{matrix}]}^{- 1} [\begin{matrix} A_{i} & M_{i} \\ M_{i}^{t} & C_{i} \end{matrix}] = [\begin{matrix} I_{3} & 0 \\ 0 & I_{1} \end{matrix}] \\ + (1 - f) {[\begin{matrix} T_{u}^{33} & T_{u}^{31} \\ T_{u}^{13} & T_{u}^{11} \end{matrix}]}^{- 1} \end{matrix}

(83b)

where T_ij are the functions of electroelastic moduli of the matrix and inclusion, as well as of the Eshelby tensor S^Y. These relationships can be considerably simplified in the elastic case. The presented mathematical formulations can be greatly simplified in some particular cases.

Piezoelectric composites with elliptical cylindrical inclusions along x₃

For more analytical expressions of Hashin–Shtrikman bounds, elliptical cylindrical piezoelectric fiber inclusions are assumed to be embedded into an elastic matrix. The resulting materials are then transversely isotropic. Using equations (69a) and (69b), analytical forms for lower and upper Hashin–Shtrikman bounds of cylindrical fibrous composites are obtained.

For the sake of convenience, the following Hill’s notations are used

\begin{matrix} Γ_{k} = \frac{S_{11}^{k} + S_{12}^{k}}{2}; k = m, i \\ Δ_{k} = \frac{S_{11}^{k} - S_{12}^{k}}{2}; k = m, i \end{matrix}

(84)

For the considered elliptic cross section of the cylinder, the aspect ratio is denoted by α = a/b, where a and b are the principal axes along x₁ and x₂ directions, respectively.

The corresponding lower and upper bounds are then given by

\begin{matrix} Γ_{l} = Γ_{m} + f_{2} \frac{S_{11}^{m} (Γ_{i} - Γ_{m}) [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (α Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ Γ_{u} = Γ_{i} + f_{1} \frac{S_{11}^{i} (Γ_{m} - Γ_{i}) [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (α Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(85a)

\begin{matrix} Δ_{l} = Δ_{m} (1 + f_{2} \frac{(1 + α) S_{11}^{m} (Δ_{i} - Δ_{m}) [(1 + α) S_{11}^{m} + 2 α f_{1} (Γ_{i} - Γ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})}) \\ Δ_{u} = Δ_{i} (1 + f_{1} \frac{(1 + α) S_{11}^{i} (Δ_{m} - Δ_{i}) [(1 + α) S_{11}^{i} + 2 α f_{2} (Γ_{m} - Γ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})}) \end{matrix}

(85b)

\begin{matrix} S_{11}^{l} = Γ_{l} + Δ_{l}, S_{12}^{l} = Γ_{l} - Δ_{l} \\ S_{11}^{u} = Γ_{u} + Δ_{u}, S_{12}^{u} = Γ_{u} - Δ_{u} \end{matrix}

(85c)

Based on these formulations, analytical bound models are obtained for the following coefficients

\begin{matrix} S_{13}^{l} = S_{13}^{m} + f_{2} \frac{S_{11}^{m} (S_{13}^{i} - S_{13}^{m}) [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (α Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ S_{13}^{u} = S_{13}^{i} + f_{1} \frac{S_{11}^{i} (S_{13}^{m} - S_{13}^{i}) [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (α Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(85d)

\begin{matrix} S_{33}^{l} = f_{1} S_{33}^{m} + f_{2} S_{33}^{i} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})] (S_{13}^{i} - S_{13}^{m})^{2}}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ S_{33}^{u} = f_{2} S_{33}^{i} + f_{1} S_{33}^{m} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})] (S_{13}^{m} - S_{13}^{i})^{2}}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(85e)

\begin{matrix} d_{31}^{l} = \frac{f_{2} S_{11}^{m} d_{31}^{i} [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (α Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ d_{31}^{u} = \frac{f_{1} S_{11}^{i} d_{31}^{m} [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (α Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(85f)

\begin{matrix} d_{32}^{l} = \frac{f_{2} S_{11}^{m} d_{31}^{i} [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 f_{1} (Δ_{m} + α S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ d_{32}^{u} = \frac{f_{1} S_{11}^{i} d_{31}^{m} [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 f_{2} (Δ_{i} + α S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(85g)

\begin{matrix} d_{33}^{l} = f_{2} (d_{33}^{i} - \frac{f_{1} d_{31}^{i} (S_{13}^{i} - S_{13}^{m}) [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})}) \\ d_{33}^{u} = f_{1} (d_{33}^{m} - \frac{f_{2} d_{31}^{m} (S_{13}^{m} - S_{13}^{i}) [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{1} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})}) \end{matrix}

(85h)

\begin{matrix} τ_{33}^{l} = f_{1} τ_{33}^{m} + f_{2} τ_{33}^{i} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})] (d_{13}^{i})^{2}}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ τ_{33}^{u} = f_{1} τ_{33}^{m} + f_{2} τ_{33}^{i} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})] (d_{13}^{m})^{2}}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(85i)

Transversely isotropic elastic materials

Bounds analytical formulations

The transversely isotropic elastic case is a particular case of the presented mathematical development. The bounds of elastic and dielectric coefficients can be easily deduced. In this case, the matrix P is reduced to

P_{k} = [\begin{matrix} A_{k} & 0 & 0 \\ 0^{t} & B_{k} & 0 \\ 0^{t} & 0^{t} & τ_{33} \end{matrix}]

(86)

where B is a diagonal matrix given by

B = [\begin{matrix} S_{44}^{h} & 0 & 0 & 0 & 0 \\ 0 & S_{55}^{h} & 0 & 0 & 0 \\ 0 & 0 & S_{66}^{h} & 0 & 0 \\ 0 & 0 & 0 & τ_{11}^{h} & 0 \\ 0 & 0 & 0 & 0 & τ_{22}^{h} \end{matrix}]

(87)

and A is a symmetric matrix given by

A = [\begin{matrix} S_{11}^{h} & S_{13}^{h} & S_{13}^{h} \\ S_{13}^{h} & S_{11}^{h} & S_{13}^{h} \\ S_{13}^{h} & S_{13}^{h} & S_{33}^{h} \end{matrix}]

(88)

Based on the previous developments, the lower and upper Hashin–Shtrikman bounds of all coefficients can be obtained numerically. For same uncoupled coefficients, simple analytical bound relationships are introduced here.

For an elastic matrix, equations (75) to (78) and (82) are considerably simplified for the lower and upper bounds of shear and dielectric moduli S₄₄, S₅₅, S₆₆, τ₁₁, τ₂₂, and τ₃₃. Explicit analytical bound models for these coefficients are given by

\begin{matrix} \frac{S_{44}^{l}}{S_{44}^{m}} = 1 + f \frac{1}{\frac{S_{44}^{m}}{(S_{44}^{i} - S_{44}^{m})} + 2 (1 - f) H_{44}} \\ \frac{S_{44}^{u}}{S_{44}^{m}} = 1 + (1 - f) \frac{1}{\frac{S_{44}^{i}}{(S_{44}^{m} - S_{44}^{i})} + 2 H_{44}} \end{matrix}

(89)

\begin{matrix} \frac{S_{55}^{l}}{S_{55}^{m}} = 1 + f \frac{1}{\frac{S_{55}^{m}}{(S_{55}^{i} - S_{55}^{m})} + 2 (1 - f) H_{55}} \\ \frac{S_{55}^{u}}{S_{55}^{m}} = 1 + (1 - f) \frac{1}{\frac{S_{55}^{i}}{(S_{55}^{m} - S_{55}^{i})} + 2 H_{55}} \end{matrix}

(90)

\begin{matrix} \frac{S_{66}^{l}}{S_{66}^{m}} = 1 + f \frac{1}{\frac{S_{66}^{m}}{(S_{66}^{i} - S_{66}^{m})} + 2 (1 - f) H_{66}} \\ \frac{S_{66}^{u}}{S_{66}^{i}} = 1 + (1 - f) \frac{1}{\frac{S_{66}^{i}}{(S_{66}^{m} - S_{66}^{i})} + 2 {fH}_{66}} \end{matrix}

(91)

\begin{matrix} \frac{τ_{11}^{l}}{τ_{11}^{m}} = 1 + f \frac{1}{\frac{τ_{11}^{m}}{(τ_{11}^{i} - τ_{11}^{m})} + (1 - f) H_{77}} \\ \frac{τ_{11}^{u}}{τ_{11}^{m}} = 1 + (1 - f) \frac{1}{\frac{τ_{11}^{i}}{(τ_{11}^{m} - τ_{11}^{i})} + H_{77}} \end{matrix}

(92)

\begin{matrix} \frac{τ_{22}^{l}}{τ_{22}^{m}} = 1 + f \frac{1}{\frac{τ_{22}^{m}}{(τ_{22}^{i} - τ_{22}^{m})} + (1 - f) H_{88}} \\ \frac{τ_{22}^{u}}{τ_{22}^{m}} = 1 + (1 - f) \frac{1}{\frac{τ_{22}^{i}}{(τ_{22}^{m} - τ_{22}^{i})} + H_{88}} \end{matrix}

(93)

\begin{matrix} \frac{τ_{33}^{l}}{τ_{33}^{m}} = 1 + f \frac{1}{\frac{τ_{33}^{m}}{(τ_{33}^{i} - τ_{33}^{m})} + (1 - f) H_{99}} \\ \frac{τ_{33}^{u}}{τ_{33}^{m}} = 1 + (1 - f) \frac{1}{\frac{τ_{33}^{i}}{(τ_{33}^{m} - τ_{33}^{i})} + H_{99}} \end{matrix}

(94)

For these moduli, the needed Eshelby tensors H components are obtained as a function of components of H^M by

\begin{matrix} H_{44} = - H_{44}^{M} + 1 / 2 \\ H_{55} = - H_{55}^{M} + 1 / 2 \\ H_{66} = - H_{66}^{M} + 1 / 2 \\ H_{77} = H_{77}^{M} \\ H_{88} = H_{88}^{M} \end{matrix}

(95)

Note that these analytical models of upper and lower bounds are given with respect to some required Eshelby tensor components. For explicit expressions, these components have to be computed analytically for some simple cases or numerically for ellipsoidal inclusions.

Elastic composite with spherical inclusions

Let us consider a composite material made of an isotropic elastic matrix containing uniformly distributed identical spherical inhomogeneities.

In this case, the main relationships (39) and (40) are reduced to

C_{m} C_{l}^{- 1} = I_{6} + f {[{(C_{i}^{- 1} - C_{m}^{- 1})}^{- 1} C_{m}^{- 1} + (1 - f) H]}^{- 1}

(96a)

C_{i} C_{u}^{- 1} = I_{6} + (1 - f) {[{(C_{m}^{- 1} - C_{i}^{- 1})}^{- 1} C_{i}^{- 1} + fH]}^{- 1}

(96b)

where C_m and C_i are, respectively, the elastic tensors of matrix and inclusion.

Using the following identities

\begin{matrix} C_{i}^{- 1} (C_{m}^{- 1} - C_{i}^{- 1})^{- 1} C_{m}^{- 1} = (C_{i} - C_{m})^{- 1} \\ C_{m}^{- 1} (C_{i}^{- 1} - C_{m}^{- 1})^{- 1} C_{i}^{- 1} = (C_{m} - C_{i})^{- 1} \end{matrix}

(97)

One gets

C_{m} C_{l}^{- 1} = I_{6} + f {[- C_{i} (C_{i} - C_{m})^{- 1} + (1 - f) H]}^{- 1}

(98)

Similarly, we have

C_{i} C_{u}^{- 1} = I_{6} + (1 - f) {[- C_{m} (C_{m} - C_{i})^{- 1} + fH]}^{- 1}

(99)

In this case, it is convenient to express Eshelby tensor coefficients in terms of elastic moduli K_m and µ_m of the matrix. For spherical inhomogeneities, the Eshelby tensor is given by Qu and Cherkaoui (2006)

H^{M} = (3 γ_{m}, 2 δ_{m})

(100)

where

\begin{matrix} γ_{m} = \frac{K_{m}}{3 K_{m} + 4 μ_{m}} = \frac{1 + ν_{m}}{9 (1 - ν_{m})} \\ δ_{m} = \frac{3 (K_{m} + 2 μ_{m})}{5 (3 K_{m} + 4 μ_{m})} = \frac{4 - 5 ν_{m}}{15 (1 - ν_{m})} \end{matrix}

(101)

in which ν_m is the Poisson ratio of the matrix phase.

From equation (29), using the notation given by Qu and Cherkaoui (2006), the needed Eshelby tensor H components are thus given by

\begin{matrix} H = (1, 1) - (3 K_{m}, 2 μ_{m}) (3 γ_{m}, 2 δ_{m}) (3 K_{m}, 2 μ_{m})^{- 1} \\ = (1 - 3 γ_{m}, 1 - 2 δ_{m}) \end{matrix}

(102)

The Hashin–Shtrikman lower bounds are thus given by

\begin{matrix} \frac{K_{m}}{K_{l}} = 1 - f \frac{1}{\frac{3 K_{m}}{3 K_{i} - 3 K_{m}} + f + (1 - f) \frac{3 K_{m}}{3 K_{m} + 4 μ_{m}}} \\ \frac{μ_{m}}{μ_{l}} = 1 - f \frac{1}{\frac{μ_{m}}{μ_{i} - μ_{m}} + f + (1 - f) \frac{6 (K_{m} + 2 μ_{m})}{5 (3 K_{m} + 4 μ_{m})}} \end{matrix}

(103)

These equations lead to the following analytical models

\begin{matrix} \frac{K_{m}}{K_{l}} = 1 - f \frac{(K_{i} - K_{m}) (3 K_{m} + 4 μ_{m})}{K_{m} (3 K_{m} + 4 μ_{m}) + f (K_{i} - K_{m}) (3 K_{m} + 4 μ_{m}) + 3 (1 - f) K_{m} (K_{i} - K_{m})} \\ \frac{μ_{m}}{μ_{l}} = 1 - f \frac{5 (μ_{i} - μ_{m}) (3 K_{m} + 4 μ_{m})}{5 μ_{m} (3 K_{m} + 4 μ_{m}) + 5 f (μ_{i} - μ_{m}) (3 K_{m} + 4 μ_{m}) + 6 (1 - f) (μ_{i} - μ_{m}) (K_{m} + 2 μ_{m})} \end{matrix}

(104)

or equivalently

\begin{matrix} \frac{K_{l}}{K_{m}} = 1 + f \frac{(K_{i} - K_{m}) (3 K_{m} + 4 μ_{m})}{K_{m} (3 K_{m} + 4 μ_{m}) + 3 (1 - f) K_{m} (K_{i} - K_{m})} \\ \frac{μ_{l}}{μ_{m}} = 1 + f \frac{5 (μ_{i} - μ_{m}) (3 K_{m} + 4 μ_{m})}{6 (1 - f) (μ_{i} - μ_{m}) (K_{m} + 2 μ_{m}) + 5 μ_{m} (3 K_{m} + 4 μ_{m})} \end{matrix}

(105)

The Hashin–Shtrikman lower bounds are finally given by the following relations

\begin{matrix} K_{l} = \frac{K_{m} (3 K_{i} + 4 μ_{m}) + 4 f μ_{m} (K_{i} - K_{m})}{(3 K_{m} + 4 μ_{m}) + 3 (1 - f) (K_{i} - K_{m})} \\ μ_{l} = \frac{6 μ_{m} μ_{i} (K_{m} + 2 μ_{m}) + μ_{m} (9 K_{m} + 8 μ_{m}) (f μ_{i} + (1 - f) μ_{m})}{μ_{m} (9 K_{m} + 8 μ_{m}) + 6 (K_{m} + 2 μ_{m}) (f μ_{m} + (1 - f) μ_{i})} \end{matrix}

(106)

Similarly, the Hashin–Shtrikman upper bounds are finally obtained

\begin{matrix} K_{u} = \frac{K_{m} (3 K_{i} + 4 μ_{i}) + 4 f μ_{i} (K_{i} - K_{m})}{(3 K_{i} + 4 μ_{i}) - 3 f (K_{i} - K_{m})} \\ μ_{u} = \frac{6 μ_{m} μ_{i} (K_{i} + 2 μ_{i}) + μ_{i} (9 K_{i} + 8 μ_{i}) (f μ_{i} + (1 - f) μ_{m})}{μ_{i} (9 K_{i} + 8 μ_{i}) + 6 (K_{i} + 2 μ_{i}) (f μ_{m} + (1 - f) μ_{i})} \end{matrix}

(107)

These last formulations are the same as those obtained by Mura (1982).

Elastic composite with elliptical cylindrical inclusions along x₃

In this case, equations (69a) and (69b) can be simplified using Hill’s notation. Using Hill’s notations, the lower and upper bounds are then given by the following expressions

\begin{matrix} Γ_{l} = Γ_{m} + f_{2} \frac{S_{11}^{m} (Γ_{i} - Γ_{m}) [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (α Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ Γ_{u} = Γ_{i} + f_{1} \frac{S_{11}^{i} (Γ_{m} - Γ_{i}) [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (α Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(108a)

\begin{matrix} Δ_{l} = Δ_{m} (1 + f_{2} \frac{(1 + α) S_{11}^{m} (Δ_{i} - Δ_{m}) [(1 + α) S_{11}^{m} + 2 α f_{1} (Γ_{i} - Γ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})}) \\ Δ_{u} = Δ_{i} (1 + f_{1} \frac{(1 + α) S_{11}^{i} (Δ_{m} - Δ_{i}) [(1 + α) S_{11}^{i} + 2 α f_{2} (Γ_{m} - Γ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})}) \end{matrix}

(108b)

\begin{matrix} S_{11}^{l} = Γ_{l} + Δ_{l}, S_{12}^{l} = Γ_{l} - Δ_{l} \\ S_{11}^{u} = Γ_{u} + Δ_{u}, S_{12}^{u} = Γ_{u} - Δ_{u} \end{matrix}

(108c)

These formulations lead to the following analytical bounds for S₁₃, S₃₃, and τ₅₅

\begin{matrix} S_{13}^{l} = S_{13}^{m} + f_{2} \frac{S_{11}^{m} (S_{13}^{i} - S_{13}^{m}) [(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (α Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})]}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ S_{13}^{u} = S_{13}^{i} + f_{1} \frac{S_{11}^{i} (S_{13}^{m} - S_{13}^{i}) [(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (α Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})]}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(109a)

\begin{matrix} S_{33}^{l} = f_{1} S_{33}^{m} + f_{2} S_{33}^{i} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})] (S_{13}^{i} - S_{13}^{m})^{2}}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ S_{33}^{u} = f_{2} S_{33}^{i} + f_{1} S_{33}^{m} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})] (S_{13}^{m} - S_{13}^{i})^{2}}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(109b)

\begin{matrix} τ_{33}^{l} = f_{1} τ_{33}^{m} + f_{2} τ_{33}^{i} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{m} S_{11}^{m} + 2 α f_{1} (Δ_{m} + S_{11}^{m}) (Δ_{i} - Δ_{m})] (d_{13}^{i})^{2}}{(1 + α)^{2} Δ_{m} S_{11}^{m} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{1} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{m} + f_{1} Δ_{m} (Γ_{i} - Γ_{m})] (Δ_{i} - Δ_{m})} \\ τ_{33}^{u} = f_{1} τ_{33}^{m} + f_{2} τ_{33}^{i} - f_{1} f_{2} \frac{[(1 + α)^{2} Δ_{i} S_{11}^{i} + 2 α f_{2} (Δ_{i} + S_{11}^{i}) (Δ_{m} - Δ_{i})] (d_{13}^{m})^{2}}{(1 + α)^{2} Δ_{i} S_{11}^{i} [f_{2} S_{11}^{m} + f_{1} S_{11}^{i}] + 2 α f_{2} [(f_{2} Γ_{m} + f_{1} Γ_{i}) S_{11}^{i} + f_{2} Δ_{i} (Γ_{m} - Γ_{i})] (Δ_{m} - Δ_{i})} \end{matrix}

(109c)

Numerical results

To illustrate the effects of piezoelectric composite microstructure based on the presented methodological approaches, the behaviors of different effective electroelastic moduli are investigated. It has to be stated that the developed mathematical approach is mainly based on the Hashin–Shtrikman model and Eshelby tensor components. For some types of inclusions such as cylindrical fibers and spherical particles, explicit analytical relationships are given. For decoupled effective coefficients, few defined piezoelectric Eshelby tensor components are required to be numerically computed for some types of inclusions.

For numerical results, two-phase piezoelectric composites are considered in which the first phase, matrix, is constituted of isotropic polymer. The second phase, inhomogeneity, is a transversely isotropic inclusion of lead zirconate titanate (PZT4). Room-temperature electroelastic constants of the two components, used for numerical results, are given in Table 1 where the epoxy matrix is assumed isotropic and the ceramic PZT4 has hexagonal symmetry. These values are obtained from the papers by Park and Sun (1995) and Chan and Unsworth (1989). For PZT4, the elastic constants are measured at constant electric field, and the dielectric constants are measured at constant strain.

Table 1.

Room-temperature elastic moduli C_ij^E (in GPa), piezoelectric coefficients e_ij (in C/m²), and relative dielectric permittivities $κ_{ii}^{ε} / κ_{0}$ of components (Chan and Unsworth, 1989; Park and Sun, 1995).

	$C_{11}^{E} (GPa)$	$C_{12}^{E} (GPa)$	$C_{13}^{E} (GPa)$	$C_{33}^{E} (GPa)$	$C_{44}^{E} (GPa)$	e₃₁ (C/m²)	e₃₁ (C/m²)	e₁₅ (C/m²)	$\frac{κ_{11}^{ε}}{κ_{0}}$	$\frac{κ_{33}^{ε}}{κ_{0}}$
PZT4	139	77.8	74.3	115	25.6	−5.2	15.1	12.7	730	635
Epoxy	8.0	4.4	4.4	8.0	1.07	0	0	0	5	5

$C_{ij}^{E}$ are measured at constant electric field and $κ_{ii}^{ε}$ are measured at constant strain.

κ₀ = 8.85 × 10⁻¹² (C²/Nm²) = permittivity of free space.

The predictions obtained by the presented Hashin–Shtrikman bound models as well as with the developed incremental DS and US bounds are presented. Numerical prediction based on the self-consistent model is also added for comparison.

The main key of the incremental bounds is the number of steps required to obtain convergence. As the steps of increment are augmented, the finite volume fraction is decreased and the lower and upper bounds are tighter. The number of steps has effect on the gap between the lower and upper bounds. Figures 1 and 2 show the effect of the number of steps on the incremental DS and US bounds related to the effective piezoelectric coefficient d₃₁. As shown in these figures, the gap between the lower and upper bounds is tighter. It is demonstrated that the two predictions $P_{dl}^{n}$ and $P_{du}^{n}$ converge to a limit called DS bound and the two predictions $P_{ul}^{n}$ and $P_{uu}^{n}$ converge to a limit called US limit bound. These two bounds, represented simultaneously in Figures 1 and 2, are considered here as new DS and US bounds. The number of steps taken into account for the incremental bounds is 50.

Calculations, done by Topolov and Bisegna (2010) and by Topolov et al. (2011), are used for comparison of the incremental DS and US bounds developed in this article. Room-temperature electroelastic moduli of the constituents, used in our predictions, are given in Table 2. These values are obtained from the books by Dantsiger et al. (1994) and Levassort et al. (1997). The comparison given in Table 3 shows the predictions obtained by the developed analytical bound relationships as well as with the incremental DS and US bounds (n = 50) with those obtained by the element-free method (EFM) and the finite element method (FEM) of Topolov and Bisegna (2010). These predictions are carried out for the piezoelectric coefficients d₃₁, d₃₂, d₃₃, and d_h of composites with 0.01 ≤ η = b/a ≤ 1 and show very good agreements for a wide range fiber sections and volume fractions.

Table 2.

Room-temperature elastic moduli C_ij^E (in GPa), piezoelectric coefficients e_ij (in C/m²), and relative dielectric permittivities $κ_{ii}^{ε} / κ_{0}$ of components (Dantsiger et al., 1994; Levassort et al., 1997).

	$C_{11}^{E} (GPa)$	$C_{12}^{E} (GPa)$	$C_{13}^{E} (GPa)$	$C_{33}^{E} (GPa)$	$C_{44}^{E} (GPa)$	e₃₁ (C/m²)	e₃₁ (C/m²)	e₁₅ (C/m²)	$\frac{κ_{11}^{ε}}{κ_{0}}$	$\frac{κ_{33}^{ε}}{κ_{0}}$
PCR-7M	133	92.0	91.0	125	22.8	−9.5	31.1	20.0	1980	1810
Araldite	7.8	4.4	4.4	7.8	1.7	0	0	0	4.0	4.0

$C_{ij}^{E}$ are measured at constant electric field and $κ_{ii}^{ε}$ are measured at constant strain.

κ₀ = 8.85 × 10⁻¹² (C²/Nm²) = permittivity of free space.

Table 3.

Effective piezoelectric predictions of d₃₁, d₃₂, d₃₃, and d_h (in pC/N) of 1–3 PCR-7M/polymer composites with various η = b/a.

η	f	d₃₁^a EFM	d₃₂^a EFM	d₃₁^a FEM	d₃₂^a FEM	d₃₁ DS	d₃₁ US	d₃₂ DS	d₃₂ US	d₃₁ HSL	d₃₁ HSU	d₃₂ HSL	d₃₂ HSU
0.01	0.01	−31,8	−25,6	−30,4	−26,4	−31,8	−32,6	−25,6	−25,6	−31,7	−32,8	−25,4	−25,7
	0.05	−120	−93,6	−111	−98,5	−120	−122	−92,4	−93,6	−120	−122	−91	−93,7
	0.10	−183	−141	−168	−150	−183	−186	−140	−141	−183	−186	−138	−142
	0.15	−222	−172	−203	−183	−222	−225	−170	−172	−221	−225	−168	−173
	0.20	−248	−194	−228	−207	−248	−250	−192	−194	−247	−251	−188	−195
	0.30	−282	−226	−260	−241	−281	−283	−224	−226	−280	−284	−220	−226
	0.50	−316	−269	−300	−285	−315	−317	−268	−269	−314	−317	−265	−269
	0.70	−333	−303	−326	−317	−332	−334	−302	−303	−332	−334	−300	−303
0.05	0.01	−30,2	−26,6	−29,9	−26,7	−29,9	−30,4	−26,5	−26,6	−29,7	−30,7	−26,4	−26,8
	0.05	−114	−97	−110	−99	−113	−115	−94	−97	−110	−128	−91	−97,2
	0.10	−175	−146	−167	−151	−175	−177	−140	−146	−172	−190	−138	−147
	0.15	−214	−176	−202	−184	−214	−216	−171	−176	−211	−227	−168	−177
	0.20	−241	−198	−227	−207	−241	−242	−193	−198	−237	−251	−190	−199
	0.30	−276	−229	−259	−241	−276	−277	−225	−229	−277	−288	−223	−230
	0.50	−313	−272	−298	−286	−312	−313	−269	−272	−312	−318	−267	−272
	0.70	−332	−303	−325	−317	−331	−332	−303	−303	−332	−336	−302	−303
0.10	0.01	−29,4	−27,1	−29,4	−27,1	−28,8	−29,5	−27,1	−27,1	−28,7	−29,7	−27	−27,3
	0.05	−111	−99	−109	−99,7	−108	−111	−97,8	−99,2	−110	−118	−93,7	−99
	0.10	−170	−149	−166	−151	−168	−171	−146	−149	−169	−179	−142	−149
	0.15	−209	−180	−201	−184	−207	−210	−177	−180	−208	−219	−173	−181
	0.20	−235	−202	−225	−208	−234	−237	−199	−202	−234	−246	−196	−203
	0.30	−271	−233	−258	−242	−270	−272	−230	−233	−270	−284	−229	−234
	0.50	−309	−274	−297	−286	−309	−310	−271	−274	−308	−318	−270	−274
	0.70	−330	−306	−323	−317	−329	−330	−304	−306	−330	−336	−304	−306
0.50	0.01	−28,3	−27,8	−28,3	−27,8	−27,8	−28,3	−27,8	−27,8	−27,7	−28,4	−27,7	−27,8
	0.05	−105	−103	−105	−103	−101	−106	−102	−103	−100	−115	−102	−104
	0.10	−160	−155	−160	−156	−155	−162	−154	−155	−150	−177	−154	−156
	0.15	−196	−188	−195	−189	−193	−198	−187	−188	−180	−216	−187	−189
	0.20	−221	−212	−220	−213	−216	−224	−210	−212	−198	−243	−211	−213
	0.30	−255	−244	−253	−246	−251	−259	−242	−244	−222	−278	−243	−245
	0.50	−296	−284	−292	−289	−295	−300	−283	−284	−245	−314	−280	−286
	0.70	−322	−313	−320	−319	−322	−324	−312	−313	−300	−332	−310	−319
1.00	0.01	−28	−28	−28	−28	−27,7	−28	−27,9	−28,1	−27,6	−28	−27,8	−28,2
	0.05	−104	−104	−104	−104	−102	−104	−102	−104	−102	−109	−102	−109
	0.10	−158	−158	−158	−158	−156	−158	−156	−156	−155	−167	−155	−167
	0.15	−192	−192	−192	−192	−190	−193	−190	−191	−187	−205	−187	−205
	0.20	−216	−216	−216	−216	−214	−218	−214	−215	−210	−231	−210	−231
	0.30	−249	−249	−249	−249	−248	−252	−248	−249	−241	−267	−241	−267
	0.50	−290	−290	−290	−290	−290	−293	−290	−291	−279	−306	−279	−306
	0.70	−317	−317	−319	−319	−317	−319	−317	−318	−307	−328	−307	−328

η	f	d₃₃^a EFM	d_h^a EFM	d₃₃^a FEM	d_h^a FEM	d₃₃ DS	d₃₃ US	d_h DS	d_h US	d₃₃ HSL	d₃₃ HSU	d_h HSL	d_h HSU
0.01	0.01	75,9	18,5	76	19,2	75,8	75,9	17,7	18,2	75,8	76	17,5	18,7
	0.05	278	64,4	278	68,5	277	278	62,4	64,6	277	278	62,3	66
	0.10	416	92	417	99	416	416	89	93	416	416	88	95
	0.15	499	105	499	113	499	499	102	107	499	499	101	110
	0.20	554	112	555	120	554	554	110	114	554	554	108	119
	0.30	623	115	623	122	623	623	114	118	623	623	113	123
	0.50	692	107	692	107	692	692	106	109	692	692	106	113
	0.70	727	91	726	83	726	727	90	92	726	727	90	94
0.05	0.01	76	19,2	76	19,4	75,9	76	19	19,5	75,9	76,1	18,6	19,8
	0.05	278	67	278	69	278	278	66	71	277	278	52,8	76
	0.10	416	95	417	99	416	416	93	101	416	416	79	106
	0.15	499	109	499	113	499	499	107	114	499	499	95	120
	0.20	554	115	555	121	554	554	114	120	554	554	104	127
	0.30	623	118	623	123	623	623	117	122	623	623	105	123
	0.50	692	107	692	108	692	692	107	111	692	692	102	113
	0.70	725	90	726	84	725	726	91	91	726	727	88	92
0.10	0.01	76,1	19,6	76,1	19,6	76	76,1	19,5	20,1	76	76,1	19,1	20,3
	0.05	278	68	278	69,3	278	278	67,8	72,2	277	278	61	73,3
	0.10	417	98	417	100	416	417	97	102	416	417	89,8	105
	0.15	499	110	499	114	499	499	109	115	499	499	99	118
	0.20	554	117	555	122	554	554	115	121	554	554	105	124
	0.30	623	119	623	123	623	623	118	123	623	623	105	124
	0.50	692	109	692	109	692	692	108	112	692	692	100	114
	0.70	727	91	726	86	726	727	91	92	726	727	85	92
0.50	0.01	76,1	20	76,1	20	76,1	76,1	20	20,5	76,1	76,1	19,9	20,7
	0.05	278	70	278	70	278	278	69	75	277	278	59	75
	0.10	417	102	417	101	416	417	100	107	416	417	84	112
	0.15	500	116	500	116	499	500	114	119	499	500	95	132
	0.20	555	122	555	122	554	555	119	128	554	555	99	145
	0.30	623	124	623	124	623	623	120	130	623	623	100	158
	0.50	692	112	692	111	692	692	108	114	692	692	92	167
	0.70	727	92	726	87	726	727	90	92	726	727	76	116
1.00	0.01	76,1	20,1	76,1	20,1	76,1	76,1	19,9	20,4	76,1	76,1	19,9	20,7
	0.05	278	70	278	70	278	278	70	74	277	278	60	73
	0.10	417	101	417	101	416	417	103	104	416	417	83	106
	0.15	500	116	500	116	499	500	116	119	499	500	90	125
	0.20	555	123	555	123	554	554	121	126	554	554	92	134
	0.30	623	125	623	125	623	623	122	127	623	623	89	141
	0.50	692	112	692	112	692	692	108	112	692	692	80	134
	0.70	727	93	726	88	726	727	90	91	725	727	71	111

Predictions are obtained based on the element-free method (EFM) and the finite element method (FEM; Topolov and Bisegna, 2010) and on the downstream (DS), the upstream (US), and the analytical expressions of Hashin–Shtrikman lower (HSL) and Hashin–Shtrikman upper (HSU) bounds developed here.

US and DS predictions are obtained by n = 50 steps.

see Topolov and Bisegna (2010).

The obtained effective coupled elastic moduli C₁₁ and C₁₃ are shown in Figures 3 and 4, respectively, for different values of the aspect ratio (α = a/b = 1/η) with respect to the volume fraction of elliptic cylindrical fibers PZT4 reinforcements. It is clearly seen that for the presented models, the aspect ratio affects significantly these effective moduli over the volume fraction of inclusion and the DS bounds and US bounds are very tight. It is noted that in the case of laminated piezocomposite (α = 100), the self-consistent predictions match well with the Hashin–Shtrikman as well as with the DS and US bounds. The piezoelectric effective moduli e₃₁ and e₃₃ are presented in Figures 5 and 6, respectively, as a function of the inclusion volume fraction. It is shown that for these coefficients, similar results tendency as for the elastic moduli are predicted by the developed models. In Figure 7, the dielectric modulus k₃₃ predictions are shown and all models predict similar results for various aspect ratios α.

Figure 3.

Effect of the aspect ratio α = a/b on the effective elastic modulus C₁₁ predicted by the explicit analytical relationships (81), downstream and upstream bounds (n = 50), and self-consistent models for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

Figure 4.

Effect of the aspect ratio α = a/b on the effective elastic modulus C₁₃ predicted by the explicit analytical relationships (81), downstream and upstream bounds (n = 50), and self-consistent models for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

Figure 5.

Effect of the aspect ratio α = a/b on the effective piezoelectric modulus e₃₁ predicted by the explicit analytical relationships (81), downstream and upstream bounds (n = 50), and self-consistent models for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

Figure 6.

Effect of the aspect ratio α = a/b on the effective piezoelectric modulus e₃₃ predicted by the explicit analytical relationships (81), downstream and upstream bounds (n = 50), and self-consistent models for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

Figure 7.

Effect of the aspect ratio α = a/b on the effective dielectric k₃₃ predicted by the explicit analytical relationships (81), downstream and upstream bounds (n = 50), and self-consistent models for PZT4 elliptic cylindrical fibers with respect to inclusion volume fractions.

The semi-analytical expressions are also used to predict the effective decoupled coefficients, and these predictions are compared to those obtained by the self-consistent and the incremental DS and US models. For ellipsoidal inclusions with spherical, ellipsoidal, and fibrous shapes, Figure 8 presents the shear modulus µ₂₃ of piezoelectric composite (PZT4-Epoxy) with respect to the volume fraction of PZT4 reinforcements. The effective piezoelectric e₁₅ and dielectric k₁₁ moduli are shown in Figures 9 and 10, respectively, as a function of the inclusion volume fraction. It is demonstrated that these coefficients exhibit strong variation with respect to the volume fraction of spherical, ellipsoidal, and fibrous shapes. A large dispersion of the obtained results is observed and particularly for large volume fractions.

Figure 8.

Effective elastic modulus µ₂₃ predicted by the explicit relationships (75), downstream and upstream bounds (n = 50), and self-consistent models with respect to PZT4 inclusion volume fraction of spherical, ellipsoidal, and fibers shapes.

Figure 9.

Effective piezoelectric modulus e₁₅ predicted by the explicit relationships (77), downstream and upstream bounds (n = 50), and self-consistent models with respect to PZT4 inclusion volume fraction of spherical, ellipsoidal, and fibers shapes.

Figure 10.

Effective normalized dielectric modulus k₁₁ predicted by the explicit relationships (64), downstream and upstream bounds (n = 50), and self-consistent models with respect to PZT4 inclusion volume fraction of spherical, ellipsoidal, and fibers shapes.

It is demonstrated that the incremental DS and the US approaches give tight bounds than the Hashin–Shtrikman model. In addition, it has to be noted that the incremental bound predictions occur inside the Hashin–Shtrikman bounds and the gap between these incremental bounds may be enhanced by increasing the number of steps.

Conclusion

In this article, analytical and semi-analytical expressions of bounds for the effective electroelastic coefficients of transversely isotropic piezoelectric inclusions in an infinite piezoelectric matrix are developed in terms of Eshelby tensor components and matrix block formulations. Closed-form expressions, based on the Hashin–Shtrikman variational principles, are obtained for the effective coupled and decoupled coefficients and only a few Eshelby tensor components are required for the considered inclusion shapes. For spherical and elliptic cylindrical inclusions, analytical expressions for the required Eshelby tensor components are known. For spheroidal inclusions, Eshelby tensor components have been explicitly provided by Mikata (2000), and for ellipsoidal ones, numerical procedures are needed. New DS and US bounds are then elaborated. It is demonstrated that the presented DS and US bounds are tighter than the Hashin–Shtrikman bounds and always inside them. The developed analytical relations are applied to two-phase composites for aligned ellipsoidal reinforcements with various aspect ratios. These relations are thus applicable to a broad range of types and geometries of composite microstructures. These simple closed-form expressions can be easily used to design composite piezoelectric materials with optimized properties. The predictions obtained using the developed analytical and semi-analytical relations show good agreements with those obtained by EFM and FEM methods as well as with the self-consistent model.

Footnotes

Acknowledgements

The authors, therefore, acknowledge with thanks DSR technical and financial support.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. 20-135-35-RG.

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Analytical and semi-analytical modeling of effective moduli bounds: Application to transversely isotropic piezoelectric materials

Abstract

Keywords

Introduction

Problem formulation

Mathematical bounds modeling

Voigt–Reuss bounds

Hashin–Shtrikman bounds

Mathematical construction of reference material

Incremental DS and US bounds

Incremental DS bounds

Incremental US bounds

Limit cases

Analytical and semi-analytical models

Piezoelectric composites with elliptical cylindrical inclusions along x3

Transversely isotropic elastic materials

Bounds analytical formulations

Elastic composite with spherical inclusions

Elastic composite with elliptical cylindrical inclusions along x3

Numerical results

Conclusion

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

References

Piezoelectric composites with elliptical cylindrical inclusions along x₃

Elastic composite with elliptical cylindrical inclusions along x₃