George Chatzigeorgiou
email: georges.chatzigeorgiou@ensam.eu

Ali Javili
email: ajavili@bilkent.edu.tr

Fodil Meraghni
email: fodil.meraghni@ensam.eu

Micromechanical method for effective piezoelectric properties and electromechanical fields in multi-coated long fiber composites

Keywords: piezoelectricity, fiber composites, multi-coated fibers, composite cylinders method

This paper proposes a micromechanical framework for identifying the macroscopic behavior of multi-coated long fiber composites, as well as the average electromechanical microscopic fields of all phases (matrix, fibers, coating layers), generated upon known macroscopic conditions. The work aims at developing a unified micromechanical approach that provides an analytical solution standing for non-coated and multi-coated long fiber composites with transversely isotropic piezoelectric behavior. The proposed method solves specific boundary value problems and utilizes the Mori-Tanaka homogenization scheme, in which the dilute strain and electric field concentration tensors are obtained analytically with the help of an extended composite cylinders method that accounts for coupled electromechanical fields. The capabilities of this homogenization strategy are illustrated with the help of numerical examples, and comparisons with known solutions from the literature for non-coated and coated fiber piezoelectric composites are provided.

Introduction

Piezoelectric materials are very attractive in applications involving the design of sensors, actuators, transducers, etc. due to their unique capability to convert electrical into mechanical energy.

Using piezoelectric ceramics, like PZT, in bulk form is not always convenient, mainly due to

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A N U S C R I P T their increased weight. To avoid such issues, an efficient solution is to combine these materials with non-piezoelectric polymers in the form of composites. These advanced composite materials opened new horizons in the development of new transducers and sensors with high strength, low thermal expansion coefficients, increased thermal conductivities and decreased dielectric constants.

During the last 30 years several models have been proposed in the literature to study the piezoelectric and the combined thermo-magneto-electro-elastic response of composites. [START_REF] Dunn | Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites[END_REF] have introduced an Eshelby-type approach by identifying appropriate Eshelby and concentration tensors for the combined electromechanical response. This technique later was extended to account for other phenomena, linear [START_REF] Li | Micromechanics of Magnetoelectroelastic Composite Materials: Average Fields and Effective Behavior[END_REF] and nonlinear [START_REF] Hossain | A multi-scale approach to model the curing process in magneto-sensitive polymeric materials[END_REF], while [START_REF] Zou | General solution for Eshelbys problem of 2D arbitrarily shaped piezoelectric inclusions[END_REF] identified Eshelby tensors for arbitrary shaped piezoelectric inclusions. [START_REF] Benveniste | On the micromechanics of fibrous piezoelectric composites[END_REF], based on the initial framework of [START_REF] Benveniste | Uniform fields and universal relations in piezoelectric composites[END_REF], studied the macroscopic response of piezoelectric composites using partially the composite cylinders method. [START_REF] Aboudi | Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites[END_REF] developed a computational method for coupled electro-magneto-thermo-elastic composites and [START_REF] Lee | Effective properties of three-phase electro-magneto-elastic composites[END_REF] have proposed numerical and Eshelby-based analytical strategies for three phase electro-magneto-elastic composites. Piezoelectric composites [START_REF] Berger | An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites[END_REF][START_REF] Berger | Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties[END_REF][START_REF] Maruccio | Computational homogenization of fibrous piezoelectric materials[END_REF], electro-magneto-thermo-elastic composites [START_REF] Bravo-Castillero | Homogenization and effective properties of periodic thermomagnetoelectroelastic composites[END_REF] and piezoelastic plates [START_REF] Kalamkarov | A new asymptotic model for a composite piezoelastic plate[END_REF] have been studied by using the periodic homogenization theory. A homogenization approach for studying piezoelectric composites with periodic and random microstructure was proposed by [START_REF] Spinelli | A general closed-form solution for the overall response of piezoelectric composites with periodic and random particulate microstructures[END_REF]. [START_REF] Sharma | On the possibility of piezoelectric nanocomposites without using piezoelectric materials[END_REF] identified a theoretical framework that describes the conditions under which non-piezoelectric materials can be used for the design of piezoelectric nanocomposites. [START_REF] Ray | Effective Properties of Carbon Nanotube and Piezoelectric Fiber Reinforced Hybrid Smart Composites[END_REF] developed a micromechanical scheme for studying piezoelectric composites with square cross section fibers, and similar technique was developed later for magneto-electro-elastic with square cross section fibers [START_REF] Pakam | Study on effective properties of 1-3-2 type magneto-electro-elastic composites[END_REF]. [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF], using Hill's interfacial operators, proposed a self consistent scheme for studying thermo-electro-elastic properties of composites with multi-coated ellipsoidal particles. [START_REF] Wang | Effective moduli of ellipsoidal particle reinforced piezoelectric composites with imperfect interfaces[END_REF] developed a micromechanical method for piezoelectric composites with imperfect interfaces between the ellipsoidal particles and the matrix phase, using the concept of equivalent particle. Piezoelectric

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A N U S C R I P T composites with imperfect interfaces have been also studied by [START_REF] Gu | Piezoelectric composites: Imperfect interface models, weak formulations and benchmark problems[END_REF][START_REF] Gu | Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings[END_REF].

The goal of this work is to develop a unified micromechanical approach aimed at providing an analytical solution that stands for non-coated and multi-coated long fiber composites with transversely isotropic piezoelectric behavior. Studying the coating between a fiber and a matrix in composites is of vital importance. A lot of nonlinear deformation mechanisms, like plasticity and/or martensite transformation, occur frequently at a small region close to the fibers and lead to an interaction with the local damage of the fiber/matrix interface [START_REF] Payandeh | Debonding initiation in a NiTi shape memory wireepoxy matrix composite. Influence of martensitic transformation[END_REF][START_REF] Payandeh | Study of the martensitic transformation in NiTiepoxy smart composite and its effect on the overall behavior[END_REF].

Developing appropriate computational tools that identify the mechanical and electric fields at the proximity of the fibers can assist in the design of more accurate damage and failure criteria for piezoelectric composites.

The developed approach is based on solving specific boundary value problems, extending the composite cylinders model of [START_REF] Hashin | The elastic moduli of fiber-reinforced materials[END_REF]. This effort can be considered as a generalization of the [START_REF] Dvorak | On transformation strains and uniform fields in multiphase elastic media[END_REF]; [START_REF] Benveniste | On the micromechanics of fibrous piezoelectric composites[END_REF] methodology, providing analytical expressions of the dilute strain-electric field coupled concentration tensors, which can be utilized in classical micromechanical techniques, like Mori-Tanaka or self consistent. The advantage of such information is that it permits to identify not only the overall response of the composite, but also the various average electromechanical fields generated at the matrix, the fiber and the coating layers for known macroscopic electromechanical conditions. For non-coated fibers the obtained dilute tensors are equivalent with those of [START_REF] Dunn | Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites[END_REF]. To the best of the authors knowledge, the only available framework in the literature that computes concentration tensors for piezoelectric composites with coated fibers is the one of [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF], but it is based on certain approximations. The approach discussed herein does not require such approximations, since the solution proposed in this work for the Eshelby's inhomogeneity problem is exact.

The organization of the manuscript is as follows: After the introduction, section 2 starts with a small recall on the piezoelectricity concepts, and then it describes the Mori-Tanaka type micromechanical framework and the general procedure for obtaining the dilute concentration tensors for composites with transversely isotropic piezoelectric material constituents (matrix, fiber, coating layers). Section 3 presents the case of non-coated fiber composites and discusses the consistency of the approach with published results from the open literature. Section 4 discusses the case of

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A N U S C R I P T coated fibers with one coating layer. Numerical examples and comparisons with existing finite element calculations and other micromechanical approaches are demonstrated in section 5. The section also includes a study in which, for given macroscopic strain and electric field and known volume fractions of the phases, all the electromechanical fields at every phase and at the overall composite are computed (Table 2). The paper finishes with a section giving some concluding remarks and future developments. For the purpose of the paper's completeness, two appendices at the end of the article provide the piezoelectric Eshelby tensor of [START_REF] Dunn | Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites[END_REF] and explain briefly the framework of [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF].

Micromechanical approach for a coated fiber/matrix piezoelectric composite

General concepts from piezoelectricity and notations

In a linear piezoelectric material, the constitutive law that describes the relation between the stress tensor σ, the electric displacement vector D, the strain tensor ε and the electric field vector E is written in the following indicial form (double indices denote summation):

σ i j = C i jkl ε kl -e i jm E m , D i = e imn ε mn + κ i j E j .
In tensorial notation, the above relations are expressed as

σ = C:ε -e• E, D = e T :ε + κ• E. (1)
In the above relations C is the fourth order elasticity tensor, κ is the second order permittivity moduli tensor and e is the third order piezoelectric moduli tensor. The strain tensor ε and electric field vector E are related with the displacement vector u and the electric potential φ respectively through the relations

ε = 1 2 gradu + [gradu] T , E = -gradφ. (2)
Moreover, the equilibrium and electrostatic equations read divσ = 0, divD = 0.

(3)

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Unless stated otherwise, the Voigt notation will be adopted in the sequel to represent second, third and fourth order tensors. The convention used in strains and stresses to replace the indices i j to a single index k is the following: Let's consider a N + 1-phase composite, consisting of a matrix, denoted as 0, and infinitely long, multi-coated, cylindrical fibers. The fibers are denoted with the index 1, while the N -1 coating layers are denoted with the indices 2,3,...,N (Figure 1). All phases are assumed to be made by linear piezoelectric materials. The phases are characterized by their piezoelectric moduli C q , e q and κ q , q = 0, 1, 2, 3, ..., N.

i j = 11 → k = 1, i j = 22 → k = 2, i j = 33 → k = 3, i j = 12 → k = 4, i j = 13 → k = 5, i j = 23 → k = 6.

Application of the Mori-Tanaka method

According to the usual micromechanics arguments, a Representative Volume Element (RVE) with total volume V is sufficient to describe the overall response of the composite. Each phase

A C C E P T E D M A N U S C R I P T Ω 0 Ω 1 r 1 Ω 2 Ω N r 2 r N Figure 2: Typical RVE of a unidirectional multi-coated fiber composite.
in the RVE occupies the space Ω q (Figure 2), has its own volume V q and volume fraction c q for q = 0, 1, 2, 3, ..., N. The classical volume summation rule states that

N q=0 V q = V, N q=0 c q = N q=0 V q V = 1.
The macroscopic strain ε, stress σ, electric field E and electric displacement D corresponding to the RVE are computed from the volume averages of their microscopic counterparts at all phases, i.e.

ε = N q=0 c q ε q , σ = N q=0 c q σ q , E = N q=0 c q E q , D = N q=0 c q D q . (4)
In these expressions ε q , σ q , E q and D q denote the average strain, stress, electric field and electric displacement in the q th phase respectively:

ε q = 1 V q Ω q ε(x) dV, σ q = 1 V q Ω q σ(x) dV, E q = 1 V q Ω q E(x) dV, D q = 1 V q Ω q D(x) dV. (5)
The constitutive law of each phase states that σ q = C q :ε q -e q • E q , D q = e T q :ε q + κ q • E q , q = 0, 1, 2, 3, ..., N.

(6)

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The goal of homogenization is to identify a similar type of constitutive law for the macroscopic quantities, i.e.

σ = C:ε -e• E, D = e T :ε + κ• E, ( 7
)
where C, κ and e are the macroscopic elasticity, permittivity and piezoelectric tensors respectively.1

The law (7) represents the overall behavior of the composite. According to the Mori-Tanaka approach, the average fields in the fiber or the coatings and the matrix phase (q = 0) are connected to each other through the dilute concentration tensors:

ε q = T mm q :ε 0 + T me q • E 0 , E q = T em q :ε 0 + T ee q • E 0 , q = 1, 2, 3, ..., N. ( 8
)
T mm q are fourth order tensors, written as 6 × 6 matrices, T me q are third order tensors, written as 6 × 3 matrices, T em q are third order tensors, written as 3 × 6 matrices and T ee q are second order tensors, written as 3 × 3 matrices. Combining ( 8) with (4) 1,3 and after some algebra, the following relations are obtained:

ε q = A mm q :ε + A me q • E, E q = A em q :ε + A ee q • E, q = 0, 1, 2, 3, ..., N, (9)
where

A mm 0 = Π mm -Π me • [Π ee ] -1 •Π em -1 , A me 0 = -A mm 0 :Π me • [Π ee ] -1 , A em 0 = -[Π ee ] -1 •Π em : A mm 0 , A ee 0 = [Π ee ] -1 -[Π ee ] -1 :Π em : A me 0 , (10)
and

A mm q = T mm q : A mm 0 + T me q • A em 0 , A me q = T mm q : A me 0 + T me q • A ee 0 , A em q = T em q : A mm 0 + T ee q • A em 0 , A ee q = T em q : A me 0 + T ee q • A ee 0 , (11)
for q = 1, 2, 3, ..., N. In the above relations,

Π mm = c 0 I + N q=1 c q T mm q , Π me = N q=1 c q T me q , Π em = N q=1 c q T em q , Π ee = c 0 I + N q=1 c q T ee q , (12) A C C E P T E D M A N U S C R I P T
while I and I, with

[I ] i jkl = 1 2 [δ ik δ jl + δ il δ jk ], [I] i j = δ i j , δ i j : Kronecker delta,
denote the symmetric fourth order and second order identity tensors respectively. Substituting ( 9) in ( 6) gives

σ q = C q : A mm q -ε q • A em q ε -e q • A ee q -C q : A me q E, D q = e T q : A mm q + κ q • A em q ε + e T q : A me q + κ q • A ee q E, ( 13
)
for q = 0, 1, 2, 3, ..., N. Implementing these results in (4) 2,4 and comparing with (7) eventually yields

C = N q=0 c q C q : A mm q -ε q • A em q , e = N q=0 c q e q • A ee q -C q : A me q , e T = N q=0 c q e T q : A mm q + κ q • A em q , κ = N q=0 c q e T q : A me q + κ q • A ee q . ( 14
)
Of course, the forms of the dilute concentration tensors T mm q , T me q , T em q and T ee q should respect the compatibility between the equations (14) 2 and (14) 3 .

It is worth mentioning that it is possible to construct one single dilute concentration tensor that combines the four tensors of the expressions (8) (see for instance [START_REF] Dunn | Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites[END_REF].

The advantage of using separate dilute tensors for each field (mechanical and electrical) and the couplings arising from them is that the scheme can be extended more naturally to account for additional mechanisms. Indeed, one can obtain dilute concentration tensors for thermomechanical response [START_REF] Benveniste | On diagonal and elastic symmetry of the approximate effective stiffness tensor oh heterogeneous media[END_REF][START_REF] A C C E P T E D M A N U S C R I P T Chatzigeorgiou | Thermomechanical Behavior of Dissipative Composite Materials[END_REF], or for inelastic response, like in the case of the Transformation Field Analysis approach [START_REF] Dvorak | On transformation strains and uniform fields in multiphase elastic media[END_REF][START_REF] Dvorak | Transformation field analysis of inelastic composite materials[END_REF].

Dilute piezoelectric concentration tensors

For the identification of the dilute concentration tensors, a single linear piezoelectric coated fiber is assumed to be embedded inside the linear piezoelectric matrix, as shown in Figure 3. The fiber occupies the space Ω 1 with volume V 1 , its coatings the spaces Ω q with volumes V q , q = 2, 3, ..., N and the matrix occupies the space Ω 0 , which is extended to far distance from the fiber. The matrix is subjected to linear displacement u 0 = ε 0 • x and linear electric potential φ 0 = -E 0 •x at far distance from the fiber (r → ∞). The interface between each phase is considered perfect. The interface between the phase q and the phase q + 1 for q = 1, ..., N -1 is denoted as ∂Ω q , while the interface between the last coating layer and the matrix is denotes as ∂Ω N .

A C C E P T E D M A N U S C R I P T u 0 = ε 0 • x, φ 0 = -E 0 • x Ω 0 Ω 1 ∂Ω N r 1 Ω 2 Ω N r 2 r N ∂Ω 1 ∂Ω 2 ∂Ω N-1
The various electromechanical fields generated at every phase q depend on the spatial position, i.e. u (q) (x), ε (q) (x), σ (q) (x), φ (q) (x), E (q) (x), D (q)

(x), ∀x ∈ Ω q . ( 15
)
In infinitely long fiber composites, it is more convenient to describe all the necessary equations in cylindrical coordinates, by describing the position vector in terms of the radius r, the angle θ and the longitudinal position z (Figure 4). For transversely isotropic piezoelectric matrix, fiber and coatings, the constitutive law (6) for each phase is written in Voigt notation as

                                                                         σ (q) rr σ (q) θθ σ (q) zz σ (q) rθ σ (q) rz σ (q) θz                                                                          =                                                                          K tr
q + µ tr q K tr qµ tr q l q 0 0 0 K tr qµ tr q K tr q + µ tr q l q 0 0 0 l q l q n q 0 0 0

0 0 0 µ tr q 0 0 0 0 0 0 µ ax q 0 0 0 0 0 0 µ ax q                                                                          •                                                                          ε (q) rr ε (q) θθ ε (q) zz 2ε (q) rθ 2ε (q) rz 2ε (q) θz                                                                          -                                                                          0 0 e q 31
0 0 e q 31 0 0 e q 33 0 0 0 e q 15 0 0 0 e q 15 0

                                                                         •                               E (q) r E (q) θ E (q) z                               , (16)
A C C E P T E D M A N U S C R I P T                               D (q) r D (q) θ D (q) z                               =                               0 0
0 0 e q 15 0 0 0 0 0 0 e q 15 e q 31 e q 31 e q 33 0 0 0

                              •                                                                         ε (q) rr ε (q) θθ ε (q) zz 2ε (q) rθ 2ε (q) rz 2ε (q) θz                                                                         +                               κ q 11 0 0 0 κ q 11 0 0 0 κ q 33                               •                               E (q) r E (q) θ E (q) z                               . ( 17
)
The strain tensor and the electric field vector at each phase are given by the expressions

ε (q) rr = ∂u (q) r ∂r , ε (q) θθ = 1 r ∂u (q) θ ∂θ + u (q) r r , ε (q) zz = ∂u (q) z ∂z , 2ε (q) rz = ∂u (q) z ∂r + ∂u (q) r ∂z , 2ε (q) θz = 1 r ∂u (q) z ∂θ + ∂u (q) θ ∂z , 2ε (q) rθ = ∂u (q) θ ∂r + 1 r ∂u (q) r ∂θ - u (q) θ r , E (q) r = - ∂φ (q) ∂r , E (q) θ = - 1 r ∂φ (q) ∂θ , E (q) z = - ∂φ (q) ∂z , (18)
while the equilibrium and electrostatic equations are written as

∂σ (q) rr ∂r + 1 r ∂σ (q) rθ ∂θ + σ (q) rr -σ (q) θθ r + ∂σ (q) rz ∂z = 0, ∂σ (q) rθ ∂r + 1 r ∂σ (q) θθ ∂θ + 2σ (q) rθ r + ∂σ (q) θz ∂z = 0, ∂σ (q) rz ∂r + 1 r ∂σ (q) θz ∂θ + σ (q) rz r + ∂σ (q) zz ∂z = 0, ∂D (q) r ∂r + 1 r ∂D (q) θ ∂θ + D (q) r r + ∂D (q) z ∂z = 0. ( 19
)
The fiber is considered to have radius r = r 1 and every coating q has external radius r q (Figure 3). The interface conditions between the phase q and the phase q + 1 for q = 1, 2, 3, ..., N -1 are

A C C E P T E D M A N U S C R I P T expressed as u (q) r (r q , θ, z) = u (q+1) r (r q , θ, z), u (q) θ (r q , θ, z) = u (q+1) θ (r q , θ, z), u (q) z (r q , θ, z) = u (q+1) z (r q , θ, z), σ (q) rr (r q , θ, z) = σ (q+1) rr (r q , θ, z), σ (q) rθ (r q , θ, z) = σ (q+1) rθ (r q , θ, z), σ (q) rz (r q , θ, z) = σ (q+1) rz (r q , θ, z), φ (q) (r q , θ, z) = φ (q+1) (r q , θ, z), D (q) r (r q , θ, z) = D (q+1) r (r q , θ, z). (20)
Additionally, the interface conditions between the N th coating and the matrix are written as

u (N) r (r N , θ, z) = u (0) r (r N , θ, z), u (N) θ (r N , θ, z) = u (0) θ (r N , θ, z), u (N) z (r N , θ, z) = u (0) z (r N , θ, z), σ (N) rr (r N , θ, z) = σ (0) rr (r N , θ, z), σ (N) rθ (r N , θ, z) = σ (0) rθ (r N , θ, z), σ (N) rz (r N , θ, z) = σ (0) rz (r N , θ, z), φ (N) (r N , θ, z) = φ (0) (r N , θ, z), D (N) r (r N , θ, z) = D (0) r (r N , θ, z). ( 21
)
Following the Eshelby's methodology, the dilute concentration tensors provide the relationship between the average strain ε q and electric field E q inside the phase q (q = 1, 2, 3, ..., N), and the strain ε 0 and electric field E 0 at the far field:

ε q = T mm q :ε 0 + T me q • E 0 , E q = T em q :ε 0 + T ee q • E 0 . (22)
The Mori-Tanaka approach assumes that the strain ε 0 and the electric field E 0 applied in the far field of this Eshelby-type problem correspond to the average strain ε 0 and the average electric field E 0 of the matrix in the RVE of Figure 2, which are used in equations ( 8). Using the divergence theorem, one obtains for the fiber the average fields

ε 1 = 1 V 1 Ω 1 ε (1) dV = 1 V 1 ∂Ω 1 1 2 [u (1) ⊗ n + n ⊗ u (1) ] dS , E 1 = 1 V 1 Ω 1 E (1) dV = 1 V 1 ∂Ω 1 [-φ (1) ]n dS , ( 23
)
where n is the unit vector of the interface ∂Ω 1 . Due to the homothetic topology of the coated fiber, the unit vector is the same at all interfaces. For all the coating layers, the average strain and electric field are defined as q) ]n dS -

ε q = 1 V q Ω q ε (q) dV = 1 V q ∂Ω q 1 2 [u (q) ⊗ n + n ⊗ u (q) ] dS - 1 V q ∂Ω q-1 1 2 [u (q-1) ⊗ n + n ⊗ u (q-1) ] dS , E q = 1 V q Ω q E (q) dV = 1 V q ∂Ω q [-φ (
∂Ω q-1 [-φ (q-1) ]n dS . ( 24
) A C C E P T E D M A N U S C R I P T
Obtaining the dilute piezoelectric tensors is not an easy task. For certain, ellipsoidal-type, forms of inclusions without coating, [START_REF] Dunn | Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites[END_REF] have computed the Eshelby tensors for the combined strain/electric field system (see Appendix A for infinitely long fibers). In the present work, the dilute concentration tensors are computed directly for long coated fiber piezoelectric composites with transversely isotropic behavior at every phase. To achieve such goal, analytical solutions are utilized of similar boundary value problems with those of the composite cylinders method proposed by [START_REF] Hashin | Thermoelastic properties of fiber composites with imperfect interface[END_REF], taking into account the effects caused by the combined presence of mechanical and electric fields. In the pure mechanical problem, similar techniques have been utilized in the literature to obtain dilute [START_REF] Benveniste | Stress fields in composites with coated inclusions[END_REF] and semi-dilute [START_REF] Chatzigeorgiou | Effective mechanical properties of aligned "fuzzy fiber" composites[END_REF] stress concentration tensors, as well as dilute strain concentration tensors [START_REF] Wang | Annular Coated Inclusion model and applications for polymer nanocomposites Part II: Cylindrical inclusions[END_REF] for coated fiber composites.

In cylindrical coordinates, the surface element in a surface of constant radius r (a vertical cylinder) is ds r = rdθdz and the surface element in a surface of constant z is ds z = rdrdθ. For an arbitrary tensor Q(r, θ, z) and a cylinder of radius r q and length 2L, the sum of surface integrals with the general form

F = 1 2Lπr q L -L 2π 0 Q(r q , θ, z)dθdz + 1 2Lπr 2 q 2π 0 r q 0 [Q(r, θ, L) -Q(r, θ, -L)] rdrdθ , (25)
is required for the computations of the average quantities ( 23) and ( 24)2 . The three normal vectors in cylindrical coordinates are expressed as

n 1 =                  cos θ sin θ 0                  , n 2 =                  -sin θ cos θ 0                  , n 3 =                  0 0 1                  . ( 26
)
The displacements of the phases are represented in matrix form as

u (q) = u (q) r n 1 + u (q) θ n 2 + u (q) z n 3 . ( 27
)
Another important point to be mentioned is that in long fiber composites with, at most, transversely isotropic phases (axis of symmetry: the direction of fibers), the dilute strain concentration tensors

A C C E P T E D M A N U S C R I P T
present transverse isotropy. In Voigt notation, they take the form

T mm q =                                                                      T mm q 11
T mm q 11 -T mm q 44

T mm q 13 0 0 0

T mm q 11 -T mm q 44 T mm q 11 T mm q 13 0 0 0 0 0 1 0 0 0 0 0 0 T mm q 44 0 0 0 0 0 0 T mm q 55 0 0 0 0 0 0 T mm q 55                                                                      , T me q =                                                                      0 0 T me q 31 0 0 T me q 31 0 0 0 0 0 0 T me q 15 0 0 0 T me q 15 0                                                                      , T em q =                             0 0 0 0 T em q 15 0 0 0 0 0 0 T em q 15 0 0 0 0 0 0                             , T ee q =                             T ee q 11 0 0 0 T ee q 11 0 0 0 1                             . ( 28
)
To obtain the unknown terms of these tensors, four types of boundary value problems should be examined. In the sequel two cases are studied: a fiber/matrix composite without coating and a fiber/matrix composite with only one coating layer.

First case: non-coated fibers

The first studied case considers non-coated piezoelectric fibers embedded in a piezoelectric matrix. This is a well examined problem in the literature and the scope of this section is to illustrate that the proposed methodology produces reliable results.

Axial shear strain and in-plane electric field conditions

For this case the following displacement vector and electric potential are applied:

u (x,y,z) 0 =                  0 0 βx                  , φ (x,y,z) 0 = -γx. (29) A C C E P T E D M A N U S C R I P T
These correspond to the following strain tensor (in classical tensorial form) and electric field vector

ε 0 =                  0 0 β/2 0 0 0 β/2 0 0                  , E 0 =                  γ 0 0                  .
In cylindrical coordinates, the displacement and electric potential are transformed to

u (r,θ,z) 0 =                  0 0 βr cos θ                  , φ (r,θ,z) 0 = -γr cos θ.
For these boundary conditions, the displacement vector and electric potential at the matrix (q = 0) and the fiber (q = 1) are given by the general expressions

u (q) z (r, θ) = βrU (q) z (r) cos θ, u (q) r = u (q) θ = 0, U (q) z (r) = Ξ (q) 1 + Ξ (q) 2 1 [r/r 1 ] 2 , φ (q) (r, θ) = -γrΦ (q) (r) cos θ, Φ (q) (r) = Ξ (q) 3 + Ξ (q) 4 1 [r/r 1 ] 2 . ( 30
)
where Ξ (q) i , i = 1, 2, 3, 4, are unknown constants. These general expressions lead to stresses and electric displacements that satisfy the equilibrium and electrostatic equations ( 19). The important stresses and electric displacements for identifying the unknown constants are σ (q) rz (r, θ) = β Σ m(q) rz (r) + γ Σ e(q) rz (r) cos θ,

D (q) r (r, θ) = β D m(q) r (r) + γ D e(q) r (r) cos θ, Σ m(q) rz (r) = µ ax q Ξ (q) 1 -Ξ (q) 2 1 [r/r 1 ] 2 , Σ e(q) rz (r) = -e q 15 Ξ (q) 3 -Ξ (q) 4 1 [r/r 1 ] 2 , D m(q) r (r) = e q 15 Ξ (q) 1 -Ξ (q) 2 1 [r/r 1 ] 2 , D e(q) r (r) = κ q 11 Ξ (q) 3 -Ξ (q) 4 1 [r/r 1 ] 2 .
The boundary conditions that should be satisfied in this boundary value problem are

u (1) z and φ (1) finite at r = 0 → Ξ (1) 2 = Ξ (1) 4 = 0, u (0) z (r → ∞, θ) = βr cos θ and φ (0) (r → ∞, θ) = -γr cos θ → Ξ (0) 1 = Ξ (0) 3 = 1 . ( 31
)
Considering these results, the interface conditions

u (1) z (r 1 , θ) = u (0) z (r 1 , θ), σ (1) rz (r 1 , θ) = σ (0) rz (r 1 , θ), φ (1) (r 1 , θ) = φ (0) (r 1 , θ), D (1) r (r 1 , θ) = D (0) r (r 1 , θ), 15 A C C E P T E D M A N U S C R I P T construct the linear system K • β I m + γ I e • [Ξ m + Ξ e ] = β F β + γ F γ , with K =                                             1 -1 0 0 µ ax 1 µ ax 0 -e 1 15 -e 0 15 0 0 1 -1 e 1 15 e 0 15 κ 1 11 κ 0 11                                             , I m =                                             1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0                                             , I e =                                             0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1                                             , Ξ m = Ξ (1) 1 Ξ (0) 2 0 0 T , Ξ e = 0 0 Ξ (1) 3 Ξ (0) 4 T , F β = 1 µ ax 0 0 e 0 15 T , F γ = 0 -e 0 15 1 κ 0 11 T .
The solution of this linear system is written in the form

Ξ m = I m • K -1 • F β + γ β K -1 • F γ , Ξ e = I e • β γ K -1 • F β + K -1 • F γ .
After some algebra, the following results are obtained:

Ξ (1) 1 = B 1 + γ β B 2 , Ξ (1) 3 = β γ B 3 + B 4 , B 1 = 2 e 0

A C C E P T E D M A N U S C R I P T

Implementing the transformations ( 25), ( 26) and ( 27) in ( 23) yields the average strain and electric field inside the fiber,

ε 1 = U (1) z (r 1 ) ε 0 → ε 1 = B 1                  0 0 β/2 0 0 0 β/2 0 0                  + B 2                  0 0 γ/2 0 0 0 γ/2 0 0                  , E 1 = Φ (1) (r 1 ) E 0 → E 1 = B 3                  β 0 0                  + B 4                  γ 0 0                 
.

Comparing these results with the expressions ( 22) and ( 28) it becomes clear that (32)

T mm 1 55 = ε 1 xz β/2 γ=0 = B 1 = 2 e 0

Transverse shear strain conditions

For this case the following displacement vector is applied:

u (x,y,z) 0 =                  βy βx 0                  , ( 33
)
which corresponds to the strain tensor (in classical tensorial form)

ε 0 =                  0 β 0 β 0 0 0 0 0                  . A C C E P T E D M A N U S C R I P T
In cylindrical coordinates, the displacement is transformed to the vector

u (r,θ,z) 0 =                  βr sin 2θ βr cos 2θ 0                  .
For these boundary conditions, the displacements at each phase are given by the general expressions

u (q) r (r, θ) = β r U (q) r (r) sin 2θ , u (q) θ (r, θ) = β r U (q) θ (r) cos 2θ , U (q) r (r) = K tr q -µ tr q 2K tr q + µ tr q [r/r 1 ] 2 Ξ (q) 1 + Ξ (q) 2 - 1 [r/r 1 ] 4 Ξ (q) 3 + K tr q + µ tr q µ tr q 1 [r/r 1 ] 2 Ξ (q)
4 ,

U (q) θ (r) = [r/r 1 ] 2 Ξ (q) 1 + Ξ (q) 2 + 1 [r/r 1 ] 4 Ξ (q) 3 + 1 [r/r 1 ] 2 Ξ (q) 4 , (34)
where the unknowns that need to be defined are the Ξ (q) i , i = 1, 2, 3, 4. These expressions lead to stresses and electric displacements that satisfy the equilibrium and electrostatic equations ( 19).

The important stresses for the calculations are given by σ (q) rr (r, θ) = β Σ (q) rr (r) sin 2θ , σ (q) rθ (r, θ) = β Σ (q) rθ (r) cos 2θ ,

Σ (q) rr (r) = 2µ tr q Ξ (q) 2 + 6µ tr q 1 [r/r 1 ] 4 Ξ (q) 3 -4K tr q 1 [r/r 1 ] 2 Ξ (q) 4 , Σ (q) rθ (r) = 6K tr q µ tr q 2K tr q + µ tr q [r/r 1 ] 2 Ξ (q) 1 + 2µ tr q Ξ (q) 2 -6µ tr q 1 [r/r 1 ] 4 Ξ (q) 3 + 2K tr q 1 [r/r 1 ] 2 Ξ (q)
4 .

The boundary conditions that should be satisfied in this boundary value problem are

u (1) r , u (1) θ finite at r = 0 → Ξ (1) 3 = Ξ (1) 4 = 0 , u (0) r (r → ∞, θ) = βr sin 2θ and u (0) θ (r → ∞, θ) = βr cos 2θ → Ξ (0) 1 = 0, Ξ (0) 2 = 1 . ( 35
)
Considering these results, the interface conditions

u (1) r (r 1 , θ) = u (0) r (r 1 , θ), u (1) θ (r 1 , θ) = u (0) θ (r 1 , θ), σ (1) rr (r 1 , θ) = σ (0) rr (r 1 , θ), σ (1) rθ (r 1 , θ) = σ (0) rθ (r 1 , θ), A C C E P T E D M A N U S C R I P T construct the linear system K •Ξ = F, with K =                                                         K tr 1 -µ tr 1 2K tr 1 + µ tr 1 1 1 - K tr 0 + µ tr 0 µ tr 0 1 1 -1 -1 0 2µ tr 1 -6µ tr 0 4K tr 0 6K tr 1 µ tr 1 2K tr 1 + µ tr 1 2µ tr 1 6µ tr 0 -2K tr 0                                                         , Ξ =                                             Ξ (1) 1 Ξ (1) 2 Ξ (0) 3 Ξ (0) 4                                             , F =                                             1 1 2µ tr 0 2µ tr 0                                             .
The solution of this linear system gives

Ξ (1) 1 = 0, Ξ (1) 2 = 2µ tr 0 K tr 0 + µ tr 0 2µ tr 0 µ tr 1 + K tr 0 µ tr 1 + µ tr 0 .
Implementing the transformations (25), ( 26) and ( 27) in (23) yields the average strain inside the fiber,

ε 1 = 1 2 U (1) r (r 1 ) + U (1) θ (r 1 ) ε 0 → ε 1 = Ξ (1) 2 ε 0 .
Comparing these results with the expressions ( 22) and ( 28) it becomes clear that

T mm 1 44 = ε 1 xy β = Ξ (1) 2 = 2µ tr 0 K tr 0 + µ tr 0 2µ tr 0 µ tr 1 + K tr 0 µ tr 1 + µ tr 0 . ( 36
)

Plane strain and axial electric field conditions

For this case the following displacement vector and electric potential are applied:

u (x,y,z) 0 =                  βx βy 0                  , φ (x,y,z) 0 = -γz. ( 37
) A C C E P T E D M A N U S C R I P T
These correspond to the following strain tensor (in classical tensorial form) and electric field vector

ε 0 =                  β 0 0 0 β 0 0 0 0                  , E 0 =                  0 0 γ                  .
In cylindrical coordinates, the displacement and electric potential are transformed to

u (r,θ,z) 0 =                  βr 0 0                  , φ (r,θ,z) 0 = -γz.
For these boundary conditions, the displacement vector and electric potential at the matrix (q = 0) and the fiber (q = 1) are given by the general expressions

u (q) r (r) = βrU (q) r (r), u (q) θ = u (q) z = 0, U (q) r (r) = Ξ (q) 1 + Ξ (q) 2 1 [r/r 1 ] 2 , φ (q) (z) = -γz, (38)
where Ξ (q) i , i = 1, 2, are unknown constants. These general expressions lead to stresses and electric displacements that satisfy the equilibrium and electrostatic equations ( 19). The important stresses for identifying the unknown constants are σ (q) rr (r) = β Σ m(q) rr (r) + γ Σ e(q) rr , Σ m(q) rr (r) = 2 K tr q Ξ (q) 1µ tr q Ξ (q) 2 1 [r/r 1 ] 2 , Σ e(q) rr = -e (q) 31 .

The boundary conditions that should be satisfied in this boundary value problem are

u (1) r finite at r = 0 → D (1) 2 = 0 , u (0) r (r → ∞) = βr → D (0) 1 = 1 . ( 39
)
Considering these results, the interface conditions u (1) r (r 1 ) = u (0) r (r 1 ), σ (1) rr (r 1 ) = σ (0) rr (r 1 ), construct the linear system

K •Ξ = F β + γ β F γ , A C C E P T E D M A N U S C R I P T with K =                 1 -1 2K tr 1 2µ tr 0                 , Ξ =                 Ξ (1) 1 Ξ (0) 2                 , F β =                 1 2K tr 0                 , F γ =                 0 e 1 31 -e 0 31                 .
The solution of this linear system gives

Ξ (1) 1 = B 1 + γ β B 2 , B 1 = K tr 0 + µ tr 0 K tr 1 + µ tr 0 , B 2 = e 1 31 -e 0 31 2K tr 1 + 2µ tr 0 . ( 40
)
Implementing the transformations ( 25), ( 26) and ( 27) in ( 23) yields the average strain inside the fiber,

ε 1 = U (1) r (r 1 ) ε 0 → ε 1 = B 1                  β 0 0 0 β 0 0 0 0                  + B 2                  γ 0 0 0 γ 0 0 0 0                  . ( 41
)
The general form of the dilute concentration tensors ( 22) and ( 28) permits to write

ε 1 xx = [2T mm 1 11 -T mm 1 44 ] β + T me 1 31 γ.
From ( 40) and ( 41), it becomes clear that

T mm 1 11 = 1 2 K tr 0 + µ tr 0 K tr 1 + µ tr 0 + T mm

Hydrostatic strain conditions

For this case the following displacement vector is applied:

u (x,y,z) 0 =                  βx βy βz                  , ( 43
)
which corresponds to the strain tensor (in classical tensorial form)

ε 0 =                  β 0 0 0 β 0 0 0 β                  . A C C E P T E D M A N U S C R I P T
In cylindrical coordinates, the displacement is transformed to the vector

u (r,θ,z) 0 =                  βr 0 βz                  .
For these boundary conditions, the displacement vector and electric potential at the matrix (q = 0) and the fiber (q = 1) are given by the general expressions

u (q) r (r) = βrU (q) r (r), u (q) θ = 0, u (q) z (z) = βz, U (q) r (r) = Ξ (q) 1 + Ξ (q) 2 1 [r/r 1 ] 2 , ( 44
)
where Ξ (q) i , i = 1, 2, are unknown constants. These general expressions lead to stresses and electric displacements that satisfy the equilibrium and electrostatic equations ( 19). The important stresses for identifying the unknown constants are

σ (q) rr (r) = β Σ (q) rr (r), Σ (q) rr (r) = 2 K tr q Ξ (q) 1 -µ tr q Ξ (q) 2 1 [r/r 1 ] 2 + l q .
The boundary conditions that should be satisfied in this boundary value problem are

u (1) r finite at r = 0 → D (1) 2 = 0 , u (0) r (r → ∞) = βr → D (0) 1 = 1 . ( 45
)
Considering these results, the interface conditions u (1) r (r 1 ) = u (0) r (r 1 ), σ (1) rr (r 1 ) = σ (0) rr (r 1 ), construct the linear system

K •Ξ = F, with K =                 1 -1 2K tr 1 2µ tr 0                 , Ξ =                 Ξ (1) 1 Ξ (0) 2                 , F =                 1 2K tr 0 + l 0 -l 1                 . A C C E P T E D M A N U S C R I P T
The solution of this linear system gives

Ξ (1) 1 = K tr 0 + µ tr 0 K tr 1 + µ tr 0 + l 0 -l 1 2K tr 1 + 2µ tr 0 . (46)
Implementing the transformations ( 25), ( 26) and ( 27) in ( 23) yields the average strain inside the fiber,

ε 1 = U (1) r (r 1 ) ε 0 → ε 1 = Ξ (1) 1 ε 0 . ( 47
)
The general form of the dilute concentration tensors ( 22) and ( 28) permits to write

ε 1 xx = [2T mm 1 11 -T mm 1 44 + T mm 1 13 ] β.
From ( 46), ( 47) and ( 42), it becomes clear that

T mm 1 13 = l 0 -l 1 2K tr 1 + 2µ tr 0 . (48)

Macroscopic properties

The dilute concentration tensors (28), whose terms are given by ( 32), ( 36), ( 42) and ( 48), are equivalent to the global piezoelectric concentration tensor provided by Dunn and Taya (1993) (see Appendix A). Assuming that the volume fraction of fibers in the composite is c and applying the Mori-Tanaka scheme, as described in subsection 2.2, the following macroscopic properties are obtained:

K tr = cK tr 1 µ tr 0 + K tr 0 K tr 1 + µ tr 0 -cµ tr 0 cK tr 0 -cK tr 1 + K tr 1 + µ tr 0 , µ tr = µ tr 0 2µ tr 0 µ tr 1 + K tr 0 µ tr 0 -cµ tr 0 + µ tr 1 + cµ tr 1 2µ tr 0 cµ tr 0 -cµ tr 1 + µ tr 1 + K tr 0 µ tr 0 + cµ tr 0 + µ tr 1 -cµ tr 1 , l = [1 -c] K tr 1 + µ tr 0 l 0 + c K tr 0 + µ tr 0 l 1 cK tr 0 -cK tr 1 + K tr 1 + µ tr 0 , n = Z 1 cK tr 0 -cK tr 1 + K tr 1 + µ tr 0 , µ ax = W 2 W 1 , e 31 = ce 1 31 K tr 0 + µ tr 0 + [1 -c]e 0 31 K tr 1 + µ tr 0 cK tr 0 -cK tr 1 + K tr 1 + µ tr 0 , e 33 = Z 2 cK tr 0 -cK tr 1 + K tr 1 + µ tr 0 , e 15 = W 3 W 1 , κ 11 = W 4 W 1 , κ 33 = Z 3 cK tr 0 -cK tr 1 + K tr 1 + µ tr 0 , A C C E P T E D M A N U S C R I P T
where The obtained solution respects the compatibility condition between equations ( 14) 2 and ( 14) 3 .

Z 1 = c 2 [l 0 -l 1 ] 2 -K tr 0 -K tr 1 [n 0 -n 1 ] + [n 0 -cn 0 + cn 1 ] K tr 1 + µ tr 0 -c [l 0 -l 1 ] 2 + n 0 K tr 1 -K tr 0 , Z 2 = c 2 K tr 0 -K tr 1 e 1 33 -e 0 33 + [l 0 -l 1 ] e 0 31 -e 1 31 + e 0 33 -ce 0 33 + ce 1 33 K tr 1 + µ tr 0 +c [l 0 -l 1 ] e 1 31 -e 0 31 + e 0 33 K tr 0 -µ tr 0 , Z 3 = -c 2 e 0 31 -e 1 31 2 + K tr 0 -K tr 1 κ 0 33 -κ 1 33 + κ 0 33 -cκ 0 33 + cκ 1 33 K tr 1 + µ tr 0 +c e 0 31 -e 1 31 2 + κ 0 33 K tr 0 -K tr 1 , W 1 = [1 + c] 2 [e 0 15 ] 2 + 2[1 -c 2 ]e 0 15 e 1 15 + [1 -c] 2 [e 1 15 ] 2 + κ 0 11 + cκ 0 11 + κ 1 11 -cκ 1 11 µ ax 0 + cµ ax 0 + µ ax 1 -cµ ax 1 , W 2 = 2[1 -c] 2 e 0 15 e 1 15 µ ax 0 + [1 -c 2 ] [e 1 15 ] 2 + [e 0 15 ] 2 µ ax 0 + 4c[e 0 15 ] 2 µ ax 1 + κ 0 11 + cκ 0 11 + κ 1 11 -cκ 1 11 µ ax 0 -cµ ax 0 + µ ax 1 + cµ ax 1 µ ax 0 , W 3 = [1 -c 2 ] [e 0 15 ] 3 + [e 1 15 ] 2 e 0 15 + 2[1 + c] 2 [e 0 15 ] 2 + 4cκ 0 11 µ ax 0 e 1 15 +[1 -c] κ 0 11 µ ax 0 + cµ ax 0 + µ ax 1 -cµ ax 1 + κ 1 11 µ ax 0 -cµ ax 0 + µ ax 1 + cµ ax 1 e 0 15 , W 4 = 2[1 -c]
The above properties agree with the findings of [START_REF] Benveniste | On the micromechanics of fibrous piezoelectric composites[END_REF]. The transverse bulk modulus K tr is given by the same expression, while for the transverse shear modulus µ tr , Benveniste also concluded that it depends exclusively on the mechanical properties. For calculating µ tr he proposed the use of the [START_REF] Christensen | Solutions for effective shear properties in three phase sphere and cylinder models[END_REF] approach, which is a generalized self consistent method and provides quite similar results with Mori-Tanaka. The rest of the properties satisfy the universal relations [START_REF] Benveniste | Uniform fields and universal relations in piezoelectric composites[END_REF])

K tr 0 -K tr l 0 -l = K tr 1 -K tr l 1 -l = cl 1 + [1 -c]l 0 -l cn (1) + [1 -c]n (0) -n = ce 1 31 + [1 -c]e 0 31 -e 31 ce (1) 33 + [1 -c]e (0) 33 -e 33 , K tr 0 -K tr e 0 31 -e 31 = K tr 1 -K tr e 1 31 -e 31 = cl 1 + [1 -c]l 0 -l ce (1) 33 + [1 -c]e (0) 33 -e 33 = e 31 -ce 1 31 -[1 -c]e 0 31 cκ 1 33 + [1 -c]κ 0 33 -κ 33 , A C C E P T E D M A N U S C R I P T and Det                               µ ax e 15 -κ 11 µ ax 1 e 1 15 -κ 1 11 µ ax 0 e 0 15 -κ 0 11                               = 0.
There are two main advantages of the proposed methodology compared to that of [START_REF] Benveniste | On the micromechanics of fibrous piezoelectric composites[END_REF]: a) The computation of the strain and electric field concentration tensors allow the identification of the microscopic electromechanical fields for known macroscopic fields. Such information can be very valuable for detecting possible inelastic mechanisms due to high stresses in the matrix or the fiber. b) The dilute concentration tensors can be used into the Mori-Tanaka scheme for obtaining the macroscopic response of composites with more complex microstructure, for instance with two different types of fibers, or the same fibers with random orientation [START_REF] Chatzigeorgiou | Effective mechanical properties of aligned "fuzzy fiber" composites[END_REF][START_REF] Seidel | Multiscale Modeling of Multifunctional Fuzzy Fibers based on Multi-Walled Carbon Nanotubes[END_REF].

Second case: fibers with one coating layer

In this second studied case, the composite consists of coated fibers and matrix. Each fiber has radius r = r 1 and is coated with a layer that has external radius r = r 2 . In the sequel, the ratio ρ = r 1 /r 2 will be utilized. Considering as c the volume fraction of the fibers, the coating layers and the matrix volume fraction can be computed through the ratio ρ. Indeed, the volume fractions of all the phases (fiber, coating, matrix) are

c 1 = c, c 2 = c ρ 2 -c, c 0 = 1 -c 1 -c 2 = 1 - c ρ 2 . ( 49
)
The dilute concentration tensors are computed with similar boundary value problems as those discussed in the previous section. Here, only the essential different points are going to be presented.

For the sake of the procedure's simplicity, only one coating layer is considered for this section.

Nevertheless, the approach is general and the addition of more coating layers is straightforward.

A C C E P T E D M

A N U S C R I P T

Axial shear strain and in-plane electric field conditions

Similarly to the non-coated fibers, the infinite matrix is subjected to the far field displacement vector and electric potential of ( 29). For all phases (matrix, fiber, coating) the general expressions (30) for these fields introduce 12 unknowns (4 constants per phase). After eliminating four from the boundary conditions ( 31), the rest of the unknowns are identified from the interface conditions that hold between the fiber and the coating and between the coating and the matrix. The produced system of linear equations take the form

K • β I m + γ I e • [Ξ m + Ξ e ] = β F β + γ F γ , with K =                                                                                                     1 -1 -1 0 0 0 0 0 0 1 ρ 2 -ρ 2 0 0 0 0 µ ax 1 -µ ax 2 µ ax 2 0 -
                                                                                                    , A C C E P T E D M A N U S C R I P T I m =                                                         
1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

                                                         , I e =                                                          0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

                                                         , Ξ m = Ξ (1) 1 Ξ (2) 1 Ξ (2) 2 Ξ (0) 2 0 0 0 0 T , Ξ e = 0 0 0 0 Ξ (1) 3 Ξ (2) 3 Ξ (2) 4 Ξ (0) 4 T , F β = 0 1 0 µ ax 0 0 0 0 e 0 15 T , F γ = 0 0 0 -e 0 15 0 1 0 κ 0 11 T .
The solution of this linear system is expressed as

Ξ m = I m • K -1 • F β + γ β K -1 • F γ , Ξ e = I e • β γ K -1 • F β + K -1 • F γ .
Considering the dilute tensors, the important terms Ξ (1) 1 , Ξ (1) 3 , Ξ (0) 2 and Ξ (0) 4 can be written in the general form

Ξ (1) 1 = B 1 + γ β B 2 , Ξ (1) 3 = β γ B 3 + B 4 , Ξ (0) 2 = B 5 + γ β B 6 , Ξ (0) 4 = β γ B 7 + B 8 .
Implementing the transformations (25), ( 26) and ( 27) in ( 23) yields the average strain and electric field inside the fiber,

ε 1 = U (1) z (r 1 ) ε 0 → ε 1 = B 1                  0 0 β/2 0 0 0 β/2 0 0                  + B 2                  0 0 γ/2 0 0 0 γ/2 0 0                  , E 1 = Φ (1) (r 1 ) E 0 → E 1 = B 3                  β 0 0                  + B 4                  γ 0 0                  .
On the other hand, implementing the transformations (25), ( 26) and ( 27) in ( 24) and taking into account the displacement and electric potential continuity conditions yields the average strain and

A C C E P T E D M A N U S C R I P T
electric field inside the coating,

ε 2 = 1 1 -ρ 2 U (0) z (r 2 ) - ρ 2 1 -ρ 2 U (1) z (r 1 ) ε 0 → ε 2 = 1 + ρ 2 B 5 -ρ 2 B 1 1 -ρ 2                  0 0 β/2 0 0 0 β/2 0 0                  + ρ 2 B 6 -ρ 2 B 2 1 -ρ 2                  0 0 γ/2 0 0 0 γ/2 0 0                  , E 2 = 1 1 -ρ 2 Φ (0) (r 2 ) - ρ 2 1 -ρ 2 Φ (1) (r 1 ) E 0 → E 2 = ρ 2 B 7 -ρ 2 B 3 1 -ρ 2                  β 0 0                  + 1 + ρ 2 B 8 -ρ 2 B 4 1 -ρ 2                  γ 0 0                  .
Comparing these results with the expressions ( 22) and ( 28) it becomes clear that

T mm 1 55 = B 1 , T me 1 15 = B 2 , T em 1 15 = B 3 , T ee 1 11 = B 4 , T mm 2 55 = 1 + ρ 2 B 5 -ρ 2 B 1 1 -ρ 2 , T me 2 15 = ρ 2 B 6 -ρ 2 B 2 1 -ρ 2 , T em 2 15 = ρ 2 B 7 -ρ 2 B 3 1 -ρ 2 , T ee 2 11 = 1 + ρ 2 B 8 -ρ 2 B 4 1 -ρ 2 . ( 50
)

Transverse shear strain conditions

For the transverse shear term of the dilute concentration tensor, the infinite matrix is subjected to the far field displacement vector of (33). For all phases (matrix, fiber, coating) the general expressions (34) introduce 12 unknowns (4 constants per phase). After eliminating four from the boundary conditions ( 35), the rest of the unknowns are identified from the interface conditions that hold between the fiber and the coating and between the coating and the matrix. The produced system of linear equations take the form

K •Ξ = F, A C C E P T E D M A N U S C R I P T with K =                                                                                                               K tr 1 -µ tr 1 2K tr 1 + µ tr 1 1 - K tr 2 -µ tr 2 2K tr 2 + µ tr 2 -1 1 - K tr 2 + µ tr 2 µ tr 2 0 0 1 1 -1 -1 -1 -1 0 0 0 0 1 ρ 2 K tr 2 -µ tr 2 2K tr 2 + µ tr 2 1 -ρ 4 ρ 2 K tr 2 + µ tr 2 µ tr 2 ρ 4 -ρ 2 K tr 0 + µ tr 0 µ tr 0 0 0 1/ρ 2 1 ρ 4 ρ 2 -ρ 4 -ρ 2 0 2µ tr 1 0 -2µ tr 2 -6µ tr 2 4K tr 2 0 0 6K tr 1 µ tr 1 2K tr 1 + µ tr 1 2µ tr 1 - 6K tr 2 µ tr 2 2K tr 2 + µ tr 2 -2µ tr 2 6µ tr 2 -2K tr 2 0 0 0 0 0 2µ tr 2 6ρ 4 µ tr 2 -4ρ 2 K tr 2 -6ρ 4 µ tr 0 4ρ 2 K tr 0 0 0 1 ρ 2 6K tr 2 µ tr 2 2K tr 2 + µ tr 2 2µ tr 2 -6ρ 4 µ tr 2 2ρ 2 K tr 2 6ρ 4 µ tr 0 -2ρ 2 K tr 0                                                                                                               , Ξ = Ξ (1) 1 Ξ (1) 2 Ξ (2) 1 Ξ (2) 2 Ξ (2) 3 Ξ (2) 4 Ξ (0) 3 Ξ (0) 4 T , F = 0 0 1 1 0 0 2µ tr 0 2µ tr 0 T .
The solution of this linear system is expressed as

Ξ = K -1 • F.
Implementing the transformations (25), ( 26) and ( 27) in ( 23) and ( 24) and taking into account the displacement continuity conditions yields the average strain inside the fiber and the coating,

ε 1 = 1 2 U (1) r (r 1 ) + U (1) θ (r 1 ) ε 0 → ε 1 = 3K tr 1 4K tr 1 + 2µ tr 1 Ξ (1) 1 + Ξ (1) 2 ε 0 , ε 2 = 1 2[1 -ρ 2 ] U (0) r (r 2 ) + U (0) θ (r 2 ) - ρ 2 2[1 -ρ 2 ] U (1) r (r 1 ) + U (1) θ (r 1 ) ε 0 → ε 2 = 1 1 -ρ 2 1 + ρ 2 K tr 0 + 2µ tr 0 2µ tr 0 Ξ (0) 4 -ρ 2 3K tr 1 4K tr 1 + 2µ tr 1 Ξ (1) 1 -ρ 2 Ξ (1) 2 ε 0 .
Comparing these results with the expressions ( 22) and ( 28) it becomes clear that

T mm 1 44 = 3K tr 1 4K tr 1 + 2µ tr 1 Ξ (1) 1 + Ξ (1) 2 ,
T mm

2 44 = 1 1 -ρ 2 1 + ρ 2 K tr 0 + 2µ tr 0 2µ tr 0 Ξ (0) 4 -ρ 2 3K tr 1 4K tr 1 + 2µ tr 1 Ξ (1) 1 -ρ 2 Ξ (1) 2 . ( 51
)
A C C E P T E D M A N U S C R I P T

Plane strain and axial electric field conditions

Similarly to the corresponding boundary value problem of the previous section, the infinite matrix is subjected to the far field displacement vector and electric potential of (37). For all phases (matrix, fiber, coating) the general expressions (38) introduce 6 unknowns (2 constants per phase). After eliminating two from the boundary conditions (39), the rest of the unknowns are identified from the interface conditions that hold between the fiber and the coating and between the coating and the matrix. The produced system of linear equations take the form The solution of this linear system is expressed as

K •Ξ = F β + γ β F γ , with K =                                             1 -1 -1 0 0 1 ρ 2 -ρ 2 2K tr 1 -2K tr 2 2µ tr 2 0 0 2K tr 2 -2ρ 2 µ tr 2 2ρ 2 µ tr 0                                             , Ξ = Ξ (1) 1 Ξ (2) 1 Ξ (2) 2 Ξ (0)
Ξ = K -1 • F β + γ β K -1 • F γ .
Considering the dilute tensors, the important terms Ξ (1) 1 and Ξ (0) 2 can be written in the general form

Ξ (1) 1 = B 1 + γ β B 2 , Ξ (0) 2 = B 3 + γ β B 4 . A C C E P T E D M A N U S C R I P T
Implementing the transformations ( 25), ( 26) and ( 27) in ( 23) and ( 24) and taking into account the displacement continuity conditions yields the average strain inside the fiber and the coating,

ε 1 = U (1) r (r 1 ) ε 0 = B 1                  β 0 0 0 β 0 0 0 0                  + B 2                  γ 0 0 0 γ 0 0 0 0                  , ε 2 = 1 1 -ρ 2 U (0) r (r 2 ) - ρ 2 1 -ρ 2 U (1) r (r 1 ) ε 0 = 1 + ρ 2 B 3 -ρ 2 B 1 1 -ρ 2                  β 0 0 0 β 0 0 0 0                  + ρ 2 B 4 -ρ 2 B 2 1 -ρ 2                  γ 0 0 0 γ 0 0 0 0                  .
The expressions of the dilute concentration tensors ( 22) and ( 28) permit to write

ε 1 xx = [2T mm 1 11 -T mm 1 44 ] β + T me 1 31 γ, ε 2 xx = [2T mm 2 11 -T mm 2 44 ] β + T me 2 31 γ.
From these results it becomes clear that

T mm 1 11 = 1 2 B 1 + 1 2 T mm 1 44 , T me 1 31 = B 2 , T mm 2 11 = 1 + ρ 2 B 3 -ρ 2 B 1 2[1 -ρ 2 ] + 1 2 T mm 2 44 , T me 2 31 = ρ 2 B 4 -ρ 2 B 2 1 -ρ 2 . (52)

Hydrostatic strain conditions

For the remaining terms of the dilute mechanical concentration tensors, the infinite matrix is subjected to the far field displacement vector (43). For all phases (matrix, fiber, coating) the general expressions (44) introduce 6 unknowns (2 constants per phase). After eliminating two from the boundary conditions (45), the rest of the unknowns are identified from the interface conditions that hold between the fiber and the coating and between the coating and the matrix.

The produced system of linear equations take the form

K •Ξ = F, A C C E P T E D M A N U S C R I P T with K =                                             1 -1 -1 0 0 1 ρ 2 -ρ 2 2K tr 1 -2K tr 2 2µ tr 2 0 0 2K tr 2 -2ρ 2 µ tr 2 2ρ 2 µ tr 0                                             , Ξ = Ξ (1) 1 Ξ (2) 1 Ξ (2) 2 Ξ (0) 2 T , F = 0 1 l 2 -l 1 2K tr 0 + l 0 -l 2 T .
The solution of this linear system is expressed as

Ξ = K -1 • F.
Implementing the transformations ( 25), ( 26) and ( 27) in ( 23) and ( 24) and taking into account the displacement continuity conditions yields the average strain inside the fiber and the coating,

ε 1 = U (1) r (r 1 ) ε 0 = Ξ (1) 1 ε 0 , ε 2 = 1 1 -ρ 2 U (0) r (r 2 ) - ρ 2 1 -ρ 2 U (1) r (r 1 ) ε 0 = 1 + ρ 2 Ξ (0) 2 -ρ 2 Ξ (1) 1 1 -ρ 2 ε 0 .
The general form of the dilute concentration tensors ( 22) and ( 28) permits to write

ε 1 xx = [2T mm 1 11 -T mm 1 44 + T mm 1 13 ] β, ε 2 xx = [2T mm 2 11 -T mm 2 44 + T mm 2 13 ] β.
From these results it becomes clear that

T mm 1 13 = Ξ (1) 1 -2T mm 1 11 + T mm 1 44 , T mm 2 13 = 1 + ρ 2 Ξ (0) 2 -ρ 2 Ξ (1) 1 1 -ρ 2 -2T mm 2 11 + T mm 2 44 . (53)
The dilute concentration tensors (28), whose terms are given by ( 50), ( 51), ( 52) and ( 53), can be directly utilized in the Mori-Tanaka scheme, described in subsection 2.2.

Numerical examples

To illustrate the capabilities of the proposed methodology, several numerical studies are performed. In the following examples the matrix phase is considered a non-piezoelectric epoxy, the properties for these materials are taken from [START_REF] Berger | An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites[END_REF][START_REF] Berger | Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties[END_REF] and they are summarized in Table 1.

In the first example, no coating phase is considered. Several macroscopic piezoelectric properties of the fiber composite as a function of the fibers volume fraction are illustrated in Figure 5.

In the same plots the numerical results of [START_REF] Berger | Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties[END_REF]

GPa GPa GPa GPa GPa Cm -2 Cm -2 Cm -2 nCV -1 m -1 nCV -1 m -1
Table 1: Material parameters for the matrix (epoxy), the fiber (PZT-7A) and the coating layer (PZT-5). The values for the first two materials are taken from [START_REF] Berger | Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties[END_REF], while for the third material are taken from [START_REF] Berger | An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites[END_REF].

In the second parametric study, the same macroscopic properties are compared with those provided by coated fiber composites. The comparisons consider the same fiber content c, while the coating and the matrix have volume fractions that depend on the radii ratio ρ = r 1 /r 2 between the fiber and the coating/matrix interface, as given by the expressions (49). Since the coating is stronger than the matrix phase, increase in the absolute values of the electromechanical properties is expected for the coated fiber composite. In Figure 6 The obtained results show that the coupled piezoelectric parameters are the properties with the highest increase. At 60% fibers volume fraction and ρ = 0.90, both e 31 and e 15 become more than 140% higher (in absolute value) compared to the corresponding parameters of the non-coated fiber composite. For the same ρ and fibers volume fraction, the gain in mechanical properties does not exceed 55% and in electrical permittivity does not exceed 66%.

The proposed methodology and the one based on the Mori-Tanaka with the dilute concentration tensors of the [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF] theory (denoted as KBBNC in Figure 6) render exactly the same predictions for all the macroscopic properties, with the exception of the transverse shear modulus. For the µ tr there is an insignificant difference (less than 0.1%), which is due to the assumption made by [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF] in the usage of the iterfacial operators (see equation (B.7) and the discussion in Appendix B). It is worth mentioning that, in the case of multiple coatings, such assumption affects all the macroscopic properties and it can cumulate a non-negligible error in the macroscopic response. On the other hand, the methodology proposed in this work does not suffer from this issue, since the Eshelby's inhomogeneity problem is solved analytically and the dilute concentration tensors are computed through exact averaging.

As already mentioned in the previous sections, the main advantage of identifying dilute concentration tensors is that they provide information about the average microscopic fields of the various phases, when the macroscopic strain and electric field are known. To demonstrate this capability, it is considered that at a coated fiber composite with fibers volume fraction 45% and

A C C E P T E D M

A N U S C R I P T ρ = 0.9, the following uniform macroscopic strain tensor and electric field vector are applied:

ε =                                          ε xx ε yy ε zz 2ε xy 2ε xz 2ε yz                                          =                                          7 6 8 2 4 6                                          •10 -4 , E =                  E x E y E z                  =                  1 2 1                  •10 -4 GV m .
For these macroscopic conditions, the average microscopic fields at all phases (matrix, fiber, coating), as well as the macroscopic stresses and electric displacements are summarized in Table 2.

Conclusions

The article presented a micromechanical approach for identifying dilute concentration tensors for piezoelectric multi-coated, long fiber composites, which can be utilized in classical homogenization schemes, such as Mori-Tanaka and self consistent. The procedure shown here for noncoated and coated fibers with one layer is applicable to fibers with multiple coating layers, by using the boundary value problems and the analytical formalism described in sections 3 and 4.

The extensive comparisons with known analytical and computational results from the literature illustrated the method's reliability and capabilities.

The main advantage of the proposed methodology is that it solves analytically the Eshelby's inhomogeneity problem, which in turn permits the exact computation of the dilute concentration piezoelectric tensors. The described procedure is also easily extendable to account for other types of mechanisms, like thermomechanical behavior or inelastic response through a transformation field analysis type approach. In addition, the proposed approach provides new capabilities in studying piezoelectric composite materials and permits, with proper modifications, to account for imperfect mechanical [START_REF] Chatzigeorgiou | Generalized interfacial energy and size effects in composites[END_REF] and/or dielectric [START_REF] Chatzigeorgiou | Interface properties influence the effective dielectric constant of composites[END_REF] interfaces between the material constituents. where I is the extended identity tensor, which in Voigt notation is written as the 9×9 identity matrix. Substituting L with L 0 and accounting for the negative sign of the electric field, renders a T that is equivalent to the system of the four dilute concentration tensors obtained in section 3. S(L 2 ) denotes the extended Eshelby tensor, which is a function of the coating piezoelectric tensor and the shape of the fiber. For multiple coating layers, equations similar to the last one connect the strain/electric field combined vectors between two consequent layers.

Expression (B.7) is exact only when the strain and the electric field inside the fiber is uniform.

For single layer coated fibers, the four boundary value problems of section 4 show that only the case of transverse shear renders non-uniform strain inside the fiber: the term Ξ (1) 1 of equation ( 51) is generally non-zero, and thus the shear strain inside the fiber depends quadratically on the radial distance. As demonstrated in section 5, this non-uniformity has a practically insignificant effect in the macroscopic transverse shear modulus. The tensors T 1 and T 2 are the required dilute concentration tensors that can be integrated into the Mori-Tanaka micromechanical scheme.

Figure 1 :

 1 Figure 1: Unidirectional multi-coated fiber composite. The coating has N -1 distinct layers.

Figure 3 :

 3 Figure 3: Coated fiber inside a matrix, subjected to linear displacement and electric potential at far distance. All phases are piezoelectric materials.

Figure 4 :

 4 Figure 4: Cylindrical coordinate system.

  1 31 -e 2 31 e 2 31 -e 0 31 T .

  -7A and the coating layers (if they exist) are made of PZT-5. The piezoelectric

Figure 5 :

 5 Figure 5: Macroscopic piezoelectric properties of fiber composites without coating: (a) K tr , (b) µ tr , (c) µ ax , (d) e 31 , (e) e 15 and (f) κ 11 . Comparison between the proposed method and the numerical results of Berger et al. (2006) using finite element computations in periodic hexagonal unit cells.

  and average microscopic fields in a coated fiber composite with fibers volume fraction 45% andρ = 0.9.                                                                                                                               3K tr + 2µ tr 4K tr + 4µ tr K tr -2µ tr 4K tr + 4µ tr                                                                                                                               . (A.4)Considering a fiber with piezoelectric properties L 1 , the dilute concentration tensor of the combined electromechanical problem is connected with the above Eshelby tensor through the classical relationT = I + S: L -1 : [L 1 -L]

  average Z tensors in the fiber and the coating is given with the help of Hill's interfacial operators.The jump of Z across an interface between two materials is given byZ + (x) = [I + P(L 2 ): [L 1 L 2 ]] : Z -, (B.6)where I is the extended identity tensor and P(L 2 ) is the interfacial operator, which depends on the coating properties. At this point, the main approximation is that the Z -is substituted with the average value Z 1 in the fiber. With this assumption, accounting for the homothetic topology of the fiber and the coating and averaging (B.6) over the space Ω 2 yieldsZ 2 = I + S(L 2 ): L -1 2 : [L 1 -L 2 ] : Z 1 . (B.7)

  Combining (B.4), (B.5) and (B.7) and after some algebra, one obtainsZ 1 = T 1 : Z 0 , Z 2 = T 2 : Z 0 , T 1 = ρ 2 N 10 + [1ρ 2 ] N 20 : N 12 -1 , T 2 = N 12 :T 1 , (B.8)withN 10 = I + S(L 0 ): L -1 0 : [L 1 -L 0 ] , N 20 = I + S(L 0 ): L -1 0 : [L 2 -L 0 ] , N 12 = I + S(L 2 ): L -1 2 : [L 1 -L 2 ] . (B.9)

  15 e 1 15 + e 0 15 + µ ax 0 κ 1 11 + κ 0 11 e 1 15 + e 0 15 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax

			0	, B 2 = 2	e 1 15 + e 0 15	e 1 15 κ 0 11 -e 0 15 κ 1 11 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax 0	,
	B 3 = 2	e 1 15 + e 0 15	e 0 15 µ ax 1 -e 1 15 µ ax 0 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax 0	, B 4 = 2	e 0 15 e 1 15 + e 0 15 + κ 0 11 µ ax 1 + µ ax 0 e 1 15 + e 0 15 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax 0	.

  15 e 1 15 + e 0 15 + µ ax 0 κ 1 11 + κ 0 11 e 1 15 + e 0 15 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax

						,
					0
	T me 1 15 =	ε 1 xz γ/2 β=0	= B 2 = 2	e 1 15 + e 0 15	e 1 15 κ 0 11 -e 0 15 κ 1 11 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax 0	,
	T em 1 15 =	E 1 x β γ=0	= B 3 = 2	e 1 15 + e 0 15	e 0 15 µ ax 1 -e 1 15 µ ax 0 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax 0	,
	T ee 1 11 =	E 1 x γ β=0	= B 4 = 2	e 0 15 e 1 15 + e 0 15 + κ 0 11 µ ax 1 + µ ax 0 e 1 15 + e 0 15 2 + κ 1 11 + κ 0 11 µ ax 1 + µ ax 0	.

  2 e 0 15 e 1 15 κ 0 11 + [1 -c 2 ] [e 1 15 ] 2 + [e 0 15 ] 2 κ 0 11 + 4c[e 0 15 ] 2 κ 1 11 + κ 0 11 -cκ 0 11 + κ 1 11 + cκ 1 11 µ ax 0 + cµ ax 0 + µ ax 1 -cµ ax 1 κ 0 11 .

  , which solve periodic unit cell problems with hexagonal arrangement of the fibers through finite element calculations, are utilized as reference values. As shown in the Figure, except from a slight under-prediction in the transverse shear modulus at high fiber content (above 50% volume fraction), the analytical method provides excellent accuracy in identifying the macroscopic response of the composite. As expected, the absolute values of the electromechanical properties show significant increase, almost exponential, at above 50% fiber volume fraction.

						material parameters
	material	n	l	K tr	µ tr	µ ax	e 31	e 33	e 15	κ 11	κ 33
	epoxy	8	4.4	6.2	1.8	1.8	0	0	0	0.0372	0.0372
	PZT-7A	131.39 82.712 119.037 35.8 25.696	-2.12058 9.52183	9.34959	4.065	2.079
	PZT-5	111	75.2	98.2	22.8	21.1	-5.4	15.8	12.3	8.11	7.35
	units

Contrarily to the electromechanical fields, the macroscopic moduli are generally not equal to the volume averages of their microscopic counterparts. The Voigt bound, which considers such relation, often provides a poor estimate of the real response.

For infinitely long cylinder, L → ∞. To avoid infinite values, the division by volume takes care of L.

Comparison between non-coated fibers and coated fibers with ρ = 0.95 and ρ = 0.90. The dashed lines correspond to the prediction of the current methodology and the points to the predictions of the Mori-Tanaka scheme with the dilute concentration tensors according to the [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF] theory.

A C C E P T E D M A N U S C R I P T Appendix A. Eshelby tensor for infinitely long cylinder in piezoelectric material

In a piezoelectric material, the constitutive law equations ( 1) can be expressed in the global matrix form

where Σ and Z are the stress/electric displacement and strain/electric field tensors respectively, written as 9×1 vectors

and L is the symmetric global piezoelectric tensor, written as 9×9 matrix For a composite with multi-coated ellipsodal shape particles, [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF] have identified global thermo-electro-elastic concentration tensors of all phases for the self consistent method and have obtained the macroscopic thermo-electro-elastic properties. When considering the Mori-Tanaka method, these global concentration tensors can be translated to dilute concentration tensors, if one substitutes the effective medium with the matrix [START_REF] A C C E P T E D M A N U S C R I P T Chatzigeorgiou | Thermomechanical Behavior of Dissipative Composite Materials[END_REF].

This Appendix presents the essential points of the [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF] approach for obtaining dilute piezoelectric concentration tensors for a coated, infinitely long fiber with a single coating layer. The Eshelby inhomogeneity problem considers a piezoelectric coated fiber embedded inside the linear piezoelectric matrix, (Figure B.1). The fiber occupies the space Ω 1 with volume V 1 , its coating the space Ω 2 with volume V 2 , and the matrix occupies the space Ω 0 , which is extended to far distance from the fiber. Adopting the notation of Appendix A for the combined electromechan-

A C C E P T E D M A N U S C R I P T

ical constitutive response, the matrix is subjected to uniform strain/electric field extended vector Z 0 at the far field. The interface between each phase is considered perfect. The ratio between the fiber radius r 1 and the external radius the interface r 2 is denoted as ρ = r 1 /r 2 . The matrix is characterized by the piezoelectic tensor L 0 , the fiber by the tensor L 1 and the coating by the tensor

The piezoelectric properties in this Eshelby problem vary spatially and can be expressed in the differential form

where Ω denotes the total space Ω = Ω 0 ∪ Ω 1 ∪ Ω 2 and δL(x) denotes spatial alteration of the matrix properties at the region of the coating and of the fiber:

Employing Green's formalism, the simplified equation for the Z vector at any point is written [START_REF] Korringa | Theory of elastic Constants of heterogeneous media[END_REF][START_REF] Zeller | Elastic Constants of Polycrystals[END_REF])

where Γ 0 (xx ) is the piezoelectric modified Green's tensor. Averaging over the space Ω 1 ∪ Ω 2 of the fiber/coating system yields [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF])

In the above expression, Z 1 , Z 2 and Z c are the average Z tensors in the fiber, the coating and the fiber/coating combined system respectively, connected with the relation

while S(L 0 ) denotes the extended Eshelby tensor that depends on the matrix properties and the shape of the coated fiber system, which is infinitely long cylinder. Appendix A gives the analytical form of this tensor for transversely isotropic material response. The connection between the

A C C E P T E D M A N U S C R I P T

The Mori-Tanaka in this formalism follows the usual procedure for three phase composite:

considering a coated fiber composite with c 0 , c 1 and c 2 the volume fractions of the matrix, the fiber and the coating respectively (see equations ( 49)), the global concentration tensors for all phases are written as