George Chatzigeorgiou

Adil Benaarbia

Fodil Meraghni

Piezoelectric-piezomagnetic behaviour of coated long fiber composites accounting for eigenfields

Keywords: Composite Cylinders Assemblage, Coated long fiber composites, Piezoelectric-piezomagnetic materials, Electro-magneto-inelastic fields

Micromechanical approach to evaluate the electro-magneto-inelastic properties in coated long fiber composites with transversely isotropic piezoelectric-piezomagnetic behaviour

• Composite Cylinders Assemblage type of boundary conditions to obtain analytical expressions of the dilute electro-magneto-mechanical concentration tensors

• Mori-Tanaka is adapted to identify i) the overall response of the composite, and ii) the various average electro-magneto-mechanical fields of the phases • Ability of the proposed model to predict both macroscopic and average microscopic fields per phase when nonlinear fields take place 1

Introduction

Piezoelectric and piezomagnetic materials have received much attention in the last decades and are being increasingly used in various applications such as aerospace, biomedical, vibration ˚Corresponding author.

Email addresses: georges.chatzigeorgiou@ensam.eu (George Chatzigeorgiou), adil.benaarbia@ensam.eu (Adil Benaarbia), fodil.meraghni@ensam.eu (Fodil Meraghni) control and many others. These materials are extensively used as actuators, sensors and transducers. Typically, active actuator materials have the capability to convert electrical and/or magnetic energy into mechanical energy, while sensor materials provide an opposite conversion. To avoid the increased weight of conventional piezoelectric or piezomagnetic materials, these smart materials are generally combined with polymers in the form of composites. Such combination permits to develop new transducers and sensors with high strength, increased thermal conductivity, low thermal expansion and advanced electromechanical behaviour. However, the co-existence of piezoelectic and piezomagnetic coupling effects in such composite materials and the involvement of many constituents-based parameters make the modelling of their multiphysical behaviour complex. From a micromechanics point of view, special techniques are required to investigate the coupling effects and the whole performance of the composite. Such modelling tools hold the key for using these smart composites in many applications in more intelligent ways.

The modelling of the piezoelectric-piezomagnetic composites is still an open topic (e.g., [START_REF] Nan | Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases[END_REF][START_REF] Wu | Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases[END_REF][START_REF] Bishay | Micromechanical Modeling of Piezoelectric-piezomagnetic Composites Using Computational Piezo-grains (CPGs)[END_REF][START_REF] Ye | Effective mechanical properties of piezoelectricpiezomagnetic hybrid smart composites[END_REF]Kuo and Hsin, 2018, etc.).

Within the last 30 years, significant progress has been made in the development of models that study the combined thermo-electro-magneto-elastic behaviour of composites [START_REF] Tang | Micromechanical modeling of the multiphysical behavior of smart materials using the variational asymptotic method[END_REF][START_REF] Bravo-Castillero | Homogenization and effective properties of periodic thermomagnetoelectroelastic composites[END_REF][START_REF] Akbarzadeh | Thermo-Magneto-Electro-Elastic Responses of Rotating Hollow Cylinders[END_REF][START_REF] Koutsawa | Overall thermo-magneto-electro-elastic properties of multiferroics composite materials with arbitrary heterogeneities spatial distributions[END_REF]. Most of the existing models in literature were focused on the thermoelastic regime and have paid particular attention to the study of both piezoelectric and piezomagnetic coupling effects. [START_REF] Dunn | Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites[END_REF] have developed an Eshelby-type approach to investigate the electroelastic behaviour of piezoelectric composite materials by identifying appropriate Eshelby and concentration tensors. This approach has then been extended to consider electro-magneto-elastic responses [START_REF] Huang | The analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions[END_REF][START_REF] Li | Micromechanics of Magnetoelectroelastic Composite Materials: Average Fields and Effective Behavior[END_REF][START_REF] Li | Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials[END_REF]. [START_REF] Huang | Magneto-electro-elastic Eshelby tensors for a piezoelectric-piezomagnetic composite reinforced by ellipsoidal inclusions[END_REF] have identified electro-magnetoelastic Eshelby tensors for elliptical, rod, penny and ribbon shaped inclusions. [START_REF] Benveniste | Magnetoelectric effect in fibrous composites with piezoelectric and piezomagnetic phases[END_REF] has studied the electro-magnetic effect in fibrous composites with piezoelectric and piezomagnetic phases using the composite cylinders assemblage method. [START_REF] Aboudi | Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites[END_REF] has developed a homogenization micromechanical method for the prediction of the effective moduli of coupled electro-magneto-thermo-elastic composites, while [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. Moreover, the electro-magneto-thermo-elastic composites have extensively been studied using the periodic homogenization theory [START_REF] Bravo-Castillero | Homogenization of magneto-electro-elastic multilaminated materials[END_REF], 2009;[START_REF] Challagulla | Micromechanical analysis of magneto-electro-thermo-elastic composite materials with applications to multilayered structures[END_REF][START_REF] Hadjiloizi | Micromechanical modeling of piezomagneto-thermo-elastic composite structures: Part I -Theory[END_REF][START_REF] Kuo | Magnetoelectricity in coated fibrous composites of piezoelectric and piezomagnetic phases[END_REF]. Pakam and Arockiarajan (2014) have developed a micromechanical scheme for studying ferroelectric and magnetostrictive composites reinforced with square cross section fibers and subjected to high electro-magnetic loading conditions. Using Hill's interfacial operators, [START_REF] Dinzart | Magneto-electro-elastic coated inclusion problem and its application to magneticpiezoelectric composite materials[END_REF] and [START_REF] Koutsawa | Generalization of the micromechanics multi-coating approach to coupled fields composite materials with eigenfields: Effective properties[END_REF] have proposed micromechanical models based on Mori-Tanaka and self consistent schemes to investigate the thermo-electro-magneto-elastic behaviour of magneto-piezoelectric composites with multi-coated ellipsoidal particles.

This work aims at developing a unified micromechanical approach in inelasticity to analytically express the electro-magneto-elastic and inelastic concentration tensors and effective material parameters for coated long fiber composites with transversely isotropic piezoelectric-piezomagnetic behaviour. The coating between the matrix and the reinforcement is witnessing strong coupling effects with complex local behaviour. Inelastic deformation mechanisms such as plasticity and/or martensite transformation occur frequently at the coating region and strongly interact with the local damage of the matrix/reinforcement interface [START_REF] Payandeh | Debonding initiation in a NiTi shape memory wire-epoxy matrix composite. Influence of martensitic transformation[END_REF][START_REF] Payandeh | Study of the martensitic transformation in NiTi-epoxy smart composite and its effect on the overall behavior[END_REF]. To reiterate, one of the aims of this work is to develop appropriate tools to assess the electro-magneto-inelastic fields in the matrix, fibers and coating layers. The current work aims at elaborating multiscale approaches to design more accurate damage and failure criteria for piezoelectric-piezomagnetic composites.

A novel approach, adapted to the Mori-Tanaka homogenization scheme, is presented in this manuscript. The 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] to account for the inelasticity. The latter effort can be considered as a generalization of the [START_REF] Dvorak | On transformation strains and uniform fields in multiphase elastic media[END_REF] methodology, providing analytical expressions of the dilute electro-magneto-mechanical concentration tensors, which can be utilized in classical micromechanical techniques, such as Mori-Tanaka or self consistent methods. The advantage of such information is that it permits to identify not only the overall response of the composite, but also the various average electro-magneto-mechanical fields generated in the matrix, the fiber and the coating layers for known macroscopic electro-magneto-mechanical conditions.

The effect of eigenfields (inelastic strains, electric or magnetic eigenfields) on the overall response, as well as in the average response of the phases, is important for the study of nonlinear behaviour of composites. Indeed, while in the current manuscript the described boundary value problems are linear, the provided solutions can be seen as the mean to include nonlinear mechanisms per phase. The latter should be described through evolution laws of the nonlinear fields and activation criteria. Via an appropriate incremental, iterative scheme, one can use the linearized micromechanics solution provided here and adapt it to account for the constitutive laws of the phases (see for instance [START_REF] Pettermann | A thermo-elasto-plastic constitutive law for inhomogeneous materials based on an incremental Mori-Tanaka approach[END_REF] for purely mechanical multiscale analysis).

To fulfill the underlined objectives, a brief description of the Eshelby's problem for coated inhomogeneity with eigenfields at the matrix fibers and coating layers is first presented. The Eshelby problem is then solved by considering special boundary condition problems analogue to those introduced by Hashin and Rosen. Then, numerical computations are conducted and discussed in detail. These computations have permitted to demonstrate the capabilities of the developed micromechanics technique. Several conclusions, drawn from the findings, are finally put forward.

Preliminaries

For small deformations and rotations, electrostatic conditions with no electric charge and magnetostatic conditions with no current density, the strain tensor, ε, the electric field vector, e, and the magnetic field vector, h, can be expressed with the help of the displacement vector, u, the electric scalar potential, a e , and the magnetic scalar potential1 , a m , respectively, through the tensorial relations ε " 1 2 " grad u `rgrad us T ‰ , e " ´grad a e and h " ´grad a m , (1)

or the vector-type equations

ε " " ε 11 ε 22 ε 33 2ε 12 2ε 13 2ε 23 ı T " " Bu 1 Bx 1 Bu 2 Bx 2 Bu 3 Bx 3 Bu 1 Bx 2 `Bu 2 Bx 1 Bu 1 Bx 3 `Bu 3 Bx 1 Bu 2 Bx 3 `Bu 3 Bx 2  T , (2) e "
" e 1 e 2 e 3 ı T "

" ´Ba e Bx 1 ´Ba e Bx 2 ´Ba e Bx 3  T , (3) h
" " h 1 h 2 h 3 ı T " " ´Ba m Bx 1 ´Ba m Bx 2 ´Ba m Bx 3  T , (4)
where the symbol r.s T stands for the usual vector/matrix transposes in orthogonal coordinates.

On the other hand, the stress tensor, σ, the electric displacement, d, and the magnetic induction, b, written in the vector-type forms σ "

" σ 11 σ 22 σ 33 σ 12 σ 13 σ 23 ı T , d " " d 1 d 2 d 3 ı T , b " " b 1 b 2 b 3 ı T , (5)
obey the following equilibrium, electrostatic and magnetostatic equations div σ " 0, div d " 0 and div b " 0,

or, in indicial form,

Bσ 11 {Bx 1 `Bσ 12 {Bx 2 `Bσ 13 {Bx 3 " 0, Bσ 12 {Bx 1 `Bσ 22 {Bx 2 `Bσ 23 {Bx 3 " 0, Bσ 13 {Bx 1 `Bσ 23 {Bx 2 `Bσ 33 {Bx 3 " 0, Bd 1 {Bx 1 `Bd 2 {Bx 2 `Bd 3 {Bx 3 " 0, Bb 1 {Bx 1 `Bb 2 {Bx 2 `Bb 3 {Bx 3 " 0. (7)
Besides, a piezo-electro-magnetic, transversely isotropic material with eigenfields obeys the fol-lowing constitutive laws, written in matrix-type notation2

σ " L• rε ´εp s ´e• re ´ep s ´f • rh ´hp s,

d " e T • rε ´εp s `κe • re ´ep s `j• rh ´hp s, b " f T • rε ´εp s `jT • re ´ep s `κm • rh ´hp s, (8)
where the index p above a field denotes an eigenfield. Example of eigenfields are those generated by the presence of temperature (e.g. thermal expansion strain, etc.). In addition, L, κ e , κ m , e, f , j denote the elasticity, the dielectric properties, the magnetic permeabilities, the coupled electro-mechanical, the coupled magneto-mechanical and the coupled electro-magnetic tensors, respectively. For transversely isotropic behaviour with axis of symmetry parallel to the direction 3, these tensors are expressed in the following matrix-type forms , f "

L " » - - - - - - - - - - - - - - - - - - - - - - - - - K tr `µtr K tr ´µtr l 0 0 0 K tr ´µtr K tr `µtr l 0 0 0 l l n 0 0 0 0 0 0 µ tr 0 0 0 0 0 0 µ ax 0 0 0 0 0 0 µ ax fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , (9) e " » - - - - - - - - - - - - - 0
» - - - - - - - - - - - - - 0 0 f 31 0 0 f 31 0 0 f 33 0 0 0 f 15 0 0 0 f 15 0 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , (10)
The constants K tr , l, n, µ tr , µ ax , e 31 , e 33 , e 15 , f 31 , f 33 , f 15 , κ e 11 , κ e 33 , κ m 11 , κ m 33 , j 11 and j 33 are material parameters. Equation (8) can also be written in the following compact form

Σ " L• rE ´Ep s , ( 13
)
where L is the 12ˆ12 symmetric matrix given as

L " » - - - - L e f e T ´κe ´j f T ´jT ´κm fi ffi ffi ffi fl , (14)
and Σ, E, E p the 12ˆ1 vectors defined as

Σ " " σ d b ı T , E " " ε ´e ´h ı T , E p " " ε p ´ep ´hp ı T . ( 15
)
Finally, the generalized vector

U " " u 1 u 2 u 3 a e a m ı T , (16)
is also introduced.

In the sequel, when dealing with various material phases, the following notation is postulated for a quantity a: i) A superscript on the symbol of the form a pqq denotes that the quantity a is spatially dependent. ii) A subscript on the symbol of the form a q is used for constant or average value of the quantity. The index q can take the values 0, 1 or 2.

Coated long fiber composites with eigenfields

Identifying the overall properties of a composite sensitive to electro-magneto-mechanical loading conditions is the task of homogenization strategies and techniques. In this paper, the mean-field approach of the Mori-Tanaka method is followed. This method relies on two aspects:

• solving a basic (Eshelby-type) problem of a coated inhomogeneity embedded in an infinite medium, allowing to compute the so-called "interaction" or "dilute concentration" tensors.

• using this information in the level of the composite for identifying the "concentration" tensors, whose knowledge leads to link the microscopic and macroscopic fields in order to obtain the macroscopic matrix L and the macroscopic eigenfield E p of the overall composite.

Eshelby's problem for coated inhomogeneity with eigenfields in all phases

Consider a coated cylindrical inhomogeneity, embedded in an infinite medium. The medium, the inhomogeneity and the homothetic coated layer are characterized by constant generalized moduli, L 0 , L 1 and L 2 , respectively. The inhomogeneity occupies the space Ω 1 with volume V 1 , is bounded by the surface BΩ 1 and is subjected to the uniform eigenfield E p 1 . The coating layer occupies the space Ω 2 with volume V 2 , is bounded by the surfaces BΩ 1 and BΩ 2 and is subjected to the uniform eigenfield E p 2 . The medium occupies the space Ω 0 , which is extended to infinity (boundary surface BΩ 8 ), and is subjected to the uniform eigenstrain E p 0 . At far distance, a linear field

U ext " » - - - - - - - - - - u 0 1 u 0 2 u 0 3 a e 0 a m 0 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl " » - - - - - - - - - - ε 11 0 x 1 `ε12 0 x 2 `ε13 0 x 3 ε 12 0 x 1 `ε22 0 x 2 `ε23 0 x 3 ε 13 0 x 1 `ε23 0 x 2 `ε33 0 x 3 ´e1 0 x 1 ´e2 0 x 2 ´e3 0 x 3 ´h1 0 x 1 ´h2 0 x 2 ´h3 0 x 3 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , (17)
with ε i j 0 , e i 0 , h i 0 (i, j " 1, 2, 3) constant values, is applied (see Figure 1).

For this problem, which is a generalized version of the famous Eshelby inhomogeneity problem [START_REF] Eshelby | The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems[END_REF] accounting for multiphysics phenomena, the constitutive law is position dependent and reads

Σpxq " $ ' ' ' ' ' ' & ' ' ' ' ' ' % L 0 • " Epxq ´Ep 0 ‰ , x P Ω 0 , L 1 • " Epxq ´Ep 1 ‰ , x P Ω 1 , L 2 • " Epxq ´Ep 2 ‰ , x P Ω 2 . ( 18
)
Figure 1: Cross section of coated cylindrical fiber with homothetic topology inside an infinite medium. All phases have homogeneous material properties and are subjected to uniform eigenfields. Moreover, the medium is subjected to linear displacement, electric potential and magnetic potential at far distance.

The boundary conditions correspond to uniform E 0 at far distance.

The main goal of this problem is to compute the average fields inside the inhomogeneity and inside the coating layer

E 1 " 1 V 1 ż Ω 1 Epxqdx and E 2 " 1 V 2 ż Ω 2 Epxqdx, (19)
respectively, when the applied field, E 0 , at far distance and the eigenfields E p 0 , E p 1 , E p 2 are known. In other words, the purpose is to evaluate the generalized, 12ˆ12 elastic and inelastic interaction tensors, T i and T p q,i for i " 1, 2 and q " 0, 1, 2, for which the following relations hold

E i " T i • E 0 `2 ÿ q"0 T p q,i • E p q . ( 20
)
The above expression is a direct extension of the corresponding one in the original Transformation Field Analysis approach of [START_REF] Dvorak | On transformation strains and uniform fields in multiphase elastic media[END_REF]. In purely mechanical problems, the first term is the usual elastic interaction tensor of a phase, connecting the total strain in the phase i with the far field applied strain. The sum of the following terms expresses the dependence of the total strain in the phase i from the inelastic strains of all the phases in the representative volume element.

To assist the computations, the ratio φ " V 1 { rV 1 `V2 s is identified.

Proposed methodology: analytical solutions in special boundary value problems

The proposed methodology is motivated by the studied boundary value problems by [START_REF] Hashin | The elastic moduli of fiber-reinforced materials[END_REF] in their famous composite cylinders assemblage theory. The necessary modifications on these problems, discussed in the sequel, provide the exact solution for the interaction tensors. Similar technique has been utilized in various articles [START_REF] Benveniste | Stress fields in composites with coated inclusions[END_REF][START_REF] Chatzigeorgiou | Effective mechanical properties of aligned "fuzzy fiber" composites[END_REF][START_REF] Wang | Annular Coated Inclusion model and applications for polymer nanocomposites Part II: Cylindrical inclusions[END_REF]Chatzigeorgiou and Meraghni, 2019;Chatzigeorgiou et al., 2019) for studying the mechanical and piezoelectric response of long fiber composites.

Before passing to the actual boundary value problems, it is essential to express all the fields and the conservation laws in cylindrical coordinates. Inside the representative volume element the various fields generated at every phase q (q " 0, 1, 2) depend on the spatial position, i.e U pqq pxq, E pqq pxq, Σ pqq pxq, E ppqq pxq, @x P Ω q .

Expressing the Eshelby's inhomogeneity problem in cylindrical coordinates

Due to the geometry of the inhomogeneities, the problem can be transformed in cylindrical coordinates, using a system of concentric cylinders for the inhomogeneity, the coating layer and the infinite matrix (see Figure 2). In cylindrical coordinates, the axes px, y, zq are transformed to pr, θ, zq, according to the relations

x 1 " r cos θ, x 2 " r sin θ, x 3 " z.
The vectors and tensors in cylindrical coordinates are noted with an arc above the symbol. Thus, in cylindrical coordinates, the fields are written as

" ´Ba epqq Br ´1 r Ba epqq Bθ ´Ba epqq Bz  T , (23)
( h pqq " " ( h pqq r ( h pqq θ ( h pqq z ı T " " ´Ba mpqq Br ´1 r Ba mpqq Bθ ´Ba mpqq Bz  T . ( 24
)
The inhomogeneity is considered to have radius r equal to r 1 and the coating layer has external radius r 2 (see Figure 1). Using the radii of the concentric cylinders, one obtains φ " r 2 1 {r 2 2 . The continuity conditions between i) the inhomogeneity and the coating layer, and ii) the coating and the matrix are expressed through the relations Due to the transverse isotropy of all phases, the generalized modulus, L, retains the same form in cylindrical coordinates.

( u p1q s pr 1 , θ, zq " ( u p2q s pr 1 , θ, zq, s " r, θ or z, ( u p2q s pr 2 , θ, zq " ( u p0q s pr 2 , θ, zq, s " r, θ or z, a sp1q pr 1 , θ, zq " a sp2q pr 1 , θ, zq, s " e or m,
a sp2q pr 2 , θ, zq " a sp0q pr 2 , θ, zq, s " e or m, (25) and
In the following subsections, the boundary value problems are presented and the analytical form of the solution (in terms of the generalized vector ( U pqq ) is given.

Analytical boundary value problems

The four boundary value problems which allow to compute the interaction tensors are the following:

• Axial shear / in-plane electric and magnetic field

ε 13 0 ε 13 0 1 0 1 0 ε p 13 0 p 1 0 p 1 0 ε p 13 1 p 1 1 p 1 1 ε p 13 2 p 1 2 p 1 2
a m ext " ´βm r ext cos θ, (27)
where β, β e and β m are considered known. In addition, all phases are subjected to the following eigenfields (

ε ppqq " s q " 0 0 0 0 cos θ ´sin θ ı T , ( e ppqq " s e q " cos θ ´sin θ 0 ı T , ( h ppqq " s m q " cos θ ´sin θ 0 ı T , (28)
for q " 0, 1, 2 and the values of s q , s e q and s m q being known. The latter expressions correspond to uniform shear strain on the plane x 1 ´x3 and uniform electric and magnetic field in the x 1 direction (see Figure 3). The analytical solution for the displacement field, the electric potential and the magnetic potential at every r, θ and z takes the form

( u pqq r " ( u pqq θ " 0, ( u pqq z " r 2 ÿ i"1 Ξ q,i " r r 1  ξ i ´1 cos θ, a epqq " ´r 2 ÿ i"1 Ξ e q,i " r r 1  ξ i ´1 cos θ, a mpqq " ´r 2 ÿ i"1 Ξ m q,i " r r 1  ξ i ´1 cos θ, (29)
with ξ 1 " 1 and ξ 2 " ´1. The unknown coefficients Ξ q,i , Ξ e q,i and Ξ m q,i are determined from the boundary conditions, the interface relations ( 25), ( 26) and the condition that the displacements, the magnetic potential and the electric potential are finite at r " 0.

• Transverse shear strain

u z ext " 0, ( 30
)
where γ is known. All phases are also subjected to the eigenstrains (

ε ppqq " 2s q " 1 2 sin 2θ ´1 2 sin 2θ 0 cos 2θ 0 0  T , (31)
for q " 0, 1, 2 and the value of s q being known. The latter expression corresponds to uniform shear strain on the plane x 1 ´x2 (see Figure 4). The analytical solution for the displacement field at every r, θ and z takes the following form

( u pqq r " r 4 ÿ i"1 Ξ q,i ψ q,i " r r 1  ξ i ´1 sin 2θ, ( u pqq θ " r 4 ÿ i"1 Ξ q,i " r r 1  ξ i ´1 cos 2θ, ( u pqq z " 0, (32)
with

ξ 1 " 3, ψ q,1 " K q ´µq 2K q `µq , ξ 2 " 1, ψ q,2 " 1, ξ 3 " ´3, ψ q,3 " ´1, ξ 4 " ´1, ψ q,4 " K q `µq µ q . ( 33
)
In such loading case, the unknown coefficients Ξ q,i are determined from the boundary conditions, the interface relations ( 25), ( 26) and the condition that the displacements are finite at r " 0.

• Plane strain / axial electric and magnetic field

ε 11 0 ε 11 0 3 0 3 0 ε p 11 0 p 3 0 p 3 0 ε p 11 1 p 3 1 p 3 1 ε p 11 2 p 3 2 p 3 2
a m ext " ´βm z, ( 34
)
where β, γ, β e and β m are assumed known. All phases are subjected to the eigenfields (

ε ppqq " s q,β " 1 1 0 0 0 0 ı T `2s q,γ " 1 2 cos 2θ ´1 2 cos 2θ 0 ´sin 2θ 0 0  T , ( e ppqq " s e q " 0 0 1 ı T , ( h ppqq " s m q " 0 0 1 ı T , (35)
where the values of s q,β , s q,γ , s e q and s m q are known for q " 0, 1, 2. The latter expressions correspond to uniform biaxial strain state with normal components on the x 1 and x 2 directions, and uniform electric and magnetic fields in the x 3 direction. Two special cases are considered here:

uniform strain at the x 1 direction, which is obtained by setting γ " β and s q,γ " s q,β (see Figure 5),

uniform strain at the x 2 direction, which is obtained by setting γ " ´β and s q,γ " ´sq,β .

The analytical solution for the displacement field, the electric potential and the magnetic potential at every r, θ and z takes the following form

( u pqq r " r 4 ÿ i"1 Ξ q,i ψ q,i " r r 1  ξ i ´1 cos 2θ `r 2 ÿ i"1 Z q,i " r r 1  ζ i ´1 , ( u pqq θ " ´r 4 ÿ i"1 Ξ q,i " r r 1  ξ i ´1 sin 2θ, ( u pqq z " 0, a epqq " ´βe z, a mpqq " ´βm z. ( 36
)
In the above expressions, ζ 1 " 1 and ζ 2 " ´1, while ψ q,i and ξ i for i " 1, 2, 3, 4 are given by the relations described in the previous boundary value problem. Moreover, the unknown coefficients Ξ q,i , Z q,i are determined from the boundary conditions, the interface relations (25), ( 26) and the condition that the displacements are finite at r " 0.

• Hydrostatic strain / axial inelastic strain

Displacement boundary conditions at far field correspond to hydrostatic strain where

All phases are subjected to the eigenstrains ( for q " 0, 1, 2 and the value of s q being known. The latter expression corresponds to uniform inelastic strain in the x 3 direction (see Figure 6). The analytical solution for the displacement field at every r, θ and z takes the following form ( u pqq r " r

ε ppqq " s q " 0 0 1 0 0 0 ı T , (38)
2 ÿ i"1 Z q,i " r r 1  ζ i ´1 , ( u pqq θ " 0 and ( u pqq z " βz, (39)
with ζ 1 " 1 and ζ 2 " ´1. The unknown coefficients Z q,i are determined from the boundary conditions, the interface relations ( 25), ( 26) and the condition that the displacements are finite at r " 0.

Computing the interaction tensors

Solving the previously described boundary value problems, the fields ( E pqq can be computed in both the fiber and its coating. Consequently, the average values E 1 and E 2 , expressed in Cartesian coordinates, can be evaluated analytically with the help of the expressions (19). In these studies, the values of β, γ, β e , β m , s q , s q,β , s q,γ , s e q and s m q can be chosen properly in order to "construct" the interaction tensors (for instance, setting one equal to 1 and zero to the rest). The four discussed boundary value problems are sufficient for identifying all the terms of T i and T p q,i . The computations are lengthy and are omitted in the current manuscript. Similar computational procedure is described in Chatzigeorgiou and Meraghni (2019); Chatzigeorgiou et al. (2019).

It is important to note that the general forms of the elastic and inelastic interaction tensors in all phases are given by the expressions

T " » - - - - - - - T 00 T 0e T 0m T e0 T ee T em T m0 T me T mm fi ffi ffi ffi ffi ffi ffi fl , T p " » - - - - - - - T p00 T p0e T p0m T pe0 T pee T pem T pm0 T pme T pmm fi ffi ffi ffi ffi ffi ffi fl , ( 40
)
where the various submatrices are written as

T 00 " » - - - - - - - - - - - - - - - - - - - - - - 1
T p00 " » - - - - - - - - - - - - - - - - - - - - - - (41)
T m0 " » - - - - - - - 0 0 0 0 " T m xz ‰ mec 0 0 0 0 0 0 " T m xz ‰ mec 0 0 0 0 0 0 fi ffi ffi ffi ffi ffi ffi fl , ( (44)
) 45
T pm0 " » - - - - - - - 0 0 0 0 rT pm xz s mec 0 0 0 0 0 0 rT pm xz s mec 0 0 0 0 0 0 fi ffi ffi ffi ffi ffi ffi fl , (46)
T 0e " » - - - - - - - - - - - - - - - - - - - - - 0 0 rT xx s elc 0 0 rT xx s elc 0 0 0 0 0 0 rT xz s elc 0 0 0 rT xz s elc 0 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , T p0e " » - - - - - - - - - - - - - - - - - - - - - 0
T 0m " » - - - - - - - - - - - - - - - - - - - - - 0 0 rT xx s mag 0 0 rT xx s mag 0 0 0 0 0 0 rT xz s mag 0 0 0 rT xz s mag 0 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , T p0m " » - - - - - - - - - - - - - - - - - - - - - (47)
In the above formulas, the symbols rt'us mec , rt'us elc and rt'us mag denote quantities that are activated by the presence of strains, electric fields and magnetic fields, respectively. Appendix A presents the computational details for obtaining the various T terms.

Alternative approach: Hill's interfacial operators

The solution to the original problem proposed by [START_REF] Eshelby | The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems[END_REF] is valid for uncoated inhomogeneities of ellipsoidal shape inside a matrix material. In the case of coated inhomogeneity, the Eshelby approach provides information for the average strain inside the inhomogeneity/coating system. To identify the strains at each phase, one can use the concept of the interfacial operators [START_REF] Walpole | A coated inclusion in an elastic medium[END_REF][START_REF] Hill | Interfacial operators in the mechanics of composite media[END_REF]. This technique has been widely utilized in the literature for composites with mechanical [START_REF] Cherkaoui | Elastic composites with coated reinforcements: a micromechanical approach for nonhomothetic topology[END_REF][START_REF] Berbenni | Homogenization of multicoated inclusion-reinforced linear elastic composites with eigenstrains: Application to thermoelastic behavior[END_REF]Chatzigeorgiou and Meraghni, 2019), piezoelectric (Chatzigeorgiou et al., 2019), thermo-piezoelectric [START_REF] Koutsawa | Multi-coating inhomogeneities approach for the effective thermo-electro-elastic properties of piezoelectric composite materials[END_REF], piezoelectric-piezomagnetic [START_REF] Dinzart | Magneto-electro-elastic coated inclusion problem and its application to magneticpiezoelectric composite materials[END_REF], and thermo-piezoelectricpiezomagnetic [START_REF] Koutsawa | Generalization of the micromechanics multi-coating approach to coupled fields composite materials with eigenfields: Effective properties[END_REF] behaviour.

The computational steps for obtaining the elastic and interaction tensors for a composite with pure mechanical behaviour have been described in detail in Chatzigeorgiou and Meraghni (2019).

For a piezoelectric-piezomagnetic composite the procedure is similar, with the only difference that the fields have the extended form presented in section 2. It is important to point out that the use of Hill's interfacial operators implies implicitly the hypothesis that in the Eshelby problem the total fields inside the inhomogeneity are uniform. As has been illustrated in Chatzigeorgiou and Meraghni (2019), under transverse shear conditions this hypothesis is violated in long fiber composites and thus can provide inaccurate results in certain cases.

Mori-Tanaka approach

Once the tensors T i and T p q,i are identified with one of the two approaches discussed previously, one can pass to the next step and consider the composite.

A direct approach, like for instance the generalized self consistent composite cylinders method [START_REF] Christensen | Mechanics of composite materials[END_REF], is applicable for the problem under investigation. It provides the local distribution of the fields in the representative volume element, but it is quite limited, since it is designed exclusively for unidirectional fiber composites with random distribution of the fibers. The current work considers the Mori-Tanaka method for identifying the macroscopic response of the composite. A mean-field approach is more flexible in terms of microstructural characteristics; it can account for non-uniform distributions of the fibers, or the presence of different types of reinforce-ment. The restriction of the proposed methodology is that it estimates only average fields per material phase and not their actual spatial distribution, which is the case for full field approaches.

A unidirectional coated fiber composite consists of a matrix phase in which coated fibers are distributed randomly (see Figure 7). In this three-phase composite, the volume fractions of the matrix, the fibers and their coating are denoted as c q , q " 0, 1, 2. Using the ratio, φ, the following relations can be given According to the basic principle of all multiscale methods, the macroscopic fields at a macroscopic point are equal to the average of the corresponding microscopic fields over the representative volume element linked with the specific macroscopic point. In the composite considered here, the macroscopic fields, E and Σ, are given by

c 2 " ´c1 `c1 φ and c 0 " 1 ´c1 φ . ( 53
E " c 0 E 0 `c1 E 1 `c2 E 2 , Σ " c 0 Σ 0 `c1 Σ 1 `c2 Σ 2 . ( 54
)
Moreover, the constitutive law for each phase can be written in its average form

Σ r " L r •rE r ´Ep r s, r " 0, 1, 2. ( 55
)
In the Mori-Tanaka method, the far field, E 0 , and the eigenfield, E p 0 , of the expressions (20) correspond to the average field and the average eigenfield of the matrix phase of the composite.

Combining ( 20) and ( 54) 1 , and after some algebra, the following relations are derived for r "

0, 1, 2 E r " A r • E `2 ÿ q"0 A p q,r • E p q , (56)
with

A 0 " « c 0 I `2 ÿ i"1 c i T i ff ´1 , A s " T s • A 0 , A p q,0 " ´A0 • 2 ÿ i"1 c i T p q,i , A p q,s " T p q,s ´As • 2 ÿ i"1 c i T p q,i , (57)
for q " 0, 1, 2 and s " 1, 2. In the above expression, I is the 12ˆ12 identity matrix. Moreover, combining equations ( 20), ( 56), ( 57) and (54) 2 , yields

Σ " L•rE ´Ep s, (58) with
L "

2 ÿ i"0 c i L i • A i , E p " 2 ÿ q"0 A p q • E p q , A p q " L ´1 • « c q L q ´2 ÿ i"0 c i L i • A p q,i ff . ( 59
)
It is worth noting that non-uniform distributions of fibers can be accounted for by introducing appropriate orientation distribution functions in the expressions (57) and One should bear mind that the Mori-Tanaka reinforced composites with alignment may lead to non-symmetric macroscopic magneto-electro-mechanical matrix L. The symmetry issue could be potentially resolved by choosing another mean-field approach with similar structure, for example the Castañeda and Willis method [START_REF] Ponte-Castañeda | The effect of spatial distribution on the effective behavior of composite materials and cracked media[END_REF][START_REF] Giordano | Nonlinear effective properties of heterogeneous materials with ellipsoidal microstructure[END_REF].

The latter assumes a unified ellipsoidal shape for the distribution of heterogeneities, which provides a consistent homogenization scheme and ensures the symmetry of the macroscopic matrix L for non-uniform alignment of the fibers.

Numerical applications

The [START_REF] Lee | Effective properties of three-phase electro-magneto-elastic composites[END_REF]. The material parameters for the glass have been obtained from [START_REF] Dinzart | Magneto-electro-elastic coated inclusion problem and its application to magneticpiezoelectric composite materials[END_REF].

κ e 11 [C 2 /N m 2 ] 0.8¨10 ´10 112¨10 ´10 0.56¨10 ´10 1¨10 ´10 κ e 33 [C 2 /N m 2 ] 0.93¨10 ´10 126¨10 ´10 0.56¨10 ´10 1¨10 ´10 κ m 11 [N/A 2 ] -590¨10 ´6 5¨10 ´6 1¨10 ´6 1¨10 ´6 κ m 33 [N/A 2 ] 157¨10 ´6 10¨10 ´6 1¨10 ´6 1¨10 ´6 e 31 [C/m 2 ] 0 -4.4 0 0 e 33 [C/m 2 ] 0 18.6 0 0 e 15 [C/m 2 ] 0 11.6 0 0 f 31 [N/Am] 580.3 0 0 0 f 33 [N/Am] 699.7 0 0 0 f 15 [N/Am] 550 0 0 0 j 11 [C/Am] 0 0 0 0 j 33 [C/Am] 0 0 0 0

Validation

In the first subsection, the electro-magneto-elastic properties of long fibers composites with and without coating layer are computed with the proposed method and are compared with the predictions of other methods in the literature. In the first example, a composite consisting of CoFe 2 O 4 matrix and BaTiO 3 long fibers is analyzed and the results are compared with the periodic homogenization predictions provided by [START_REF] Lee | Effective properties of three-phase electro-magneto-elastic composites[END_REF]. The mechanical stiffness coefficients, the piezoelectric coupling terms and the electromagnetic coupling terms are demonstrated in Figure 8 The second example illustrates results from studies in [START_REF] Dinzart | Magneto-electro-elastic coated inclusion problem and its application to magneticpiezoelectric composite materials[END_REF]. In that paper, an Eshelby-based micromechanics technique was developed using the Hill's interfacial operators.

In their analysis, glass fibers coated with BaTiO 3 coating layer are embedded in CoFe 2 O 4 matrix.

The aspect ratio of the fibers (length to radius diameter) is 100, which is quite large and can serve for the purpose of this paper as almost long fiber. The ratio φ for the coated fibers is taken equal to 82.645%. Figure 9 illustrates the electromagnetic coupling terms of the composite as a function of the fiber volume fraction. The results of both methods are in good agreement and the small difference is mainly due to the finite value of the fibers' aspect ratio.

The above preliminary examples demonstrate that the new approach has the same accuracy with existing homogenization techniques in terms of electro-magneto-mechanical properties. The following numerical studies demonstrate the ability of the model to predict both macroscopic and average microscopic fields per phase when nonlinear fields take place.

Applied thermal strains

A typical example of inelastic deformation is the appearance of thermal strains. Consider a composite made of CoFe 2 O 4 matrix and glass fibers coated with BaTiO 3 coating layer. The composite is assumed free from external loading and only a temperature difference of 1 K is applied. The thermal expansion coefficients for the matrix, the fiber and the coating are taken equal to 10 ´5 1/K, 5 ¨10 ´6 1/K and 6.4 ¨10 ´6 1/K respectively.

According to the general principles of homogenization of composites subjected to thermomechanical processes, the microscopic and macroscopic temperature coincide [START_REF] Chatzigeorgiou | Thermomechanical Behavior of Dissipative Composite Materials[END_REF], i.e. all material phases in the RVE are subjected to the same temperature difference of 1 K.

This leads to the development of inelastic (thermal) strains in all phases:

ε p 0 " " 10 ´5 10 ´5 10 ´5 0 0 0 ı T , ε p 1 " " 5 ¨10 ´6 5 ¨10 ´6 5 ¨10 ´6 0 0 0 ı T , ε p 2 " " 6.4 ¨10 ´6 6.4 ¨10 ´6 6.4 ¨10 ´6 0 0 0 ı T .
Figure 10 illustrates the non zero terms of the macroscopic stress tensor, the electric displacement vector and the magnetic induction vector that are generated to the thermal strains function of the volume fraction. Considering the coating, two different ratios φ are considered: φ " 50% and φ " 80%. From these results one observes that the stresses and the magnetic induction increase with the increase of c 1 , while the electric displacement initially has a small increase and then decreases as the fiber volume fraction obtains high values. -5 φ " 80%.

-4 -3 -2 -1 0 1 σ 11 [MPa] σ 22 [MPa] σ 33 [MPa] 3 [10 ´5 C/m 2 ] 3 [10 ´2 N/Am] Fiber volume fraction [%] 0 20 40 60 80 Macroscopic fields -7 -6 -5 -4 -3 -2 -1 0 1 σ 11 [MPa] σ 22 [MPa] σ 33 [MPa] 3 [10 ´5 C/m 2 ] 3 [10 ´2 N/Am] (a) (b)

0.00

Microscopic fields (c 1 " 20%, φ " 50%) The average microscopic fields per phase for fiber volume fraction 20% and ratio φ " 50% are demonstrated in Figure 11. As it can be observed, the electric displacement for the matrix and the fiber are zero. This phenomenon is explained by the structure of the interaction tensors (40) and the constitutive relations (8). The terms T e0 and T pe0 are linked with the development of electric fields inside the fiber and the coating due to the presence of mechanical strains. Similarly, the terms T m0

σ 11 [MPa] 3 [10 ´5 C/m 2 ] 3 [10 ´2 N/Am]
and T pm0 are linked with the development of magnetic fields inside the fiber and the coating due to the presence of mechanical strains. As it can be seen by the expressions ( 43), ( 44), ( 45) and ( 46) for these terms, elastic or inelastic normal strains cannot produce electric or magnetic field in the phases. Thus, according to the constitutive law relations (8), the development of electric displacement can appear only in phases non zero coupling tensor as it the case for the (BaTiO ). The glass and the CoFe 2 4 have zero piezoelectric coefficients, and thus no electric field can be generated in these materials under only thermal strain conditions. For completely analogous reasons, magnetic induction can only be generated in the matrix (CoFe 2 O 4 )

and not in the fiber or the coating. Contrarily to the microscopic scale, electric displacement and magnetic induction parallel to the fiber axis direction are generated at the macroscopic response, as it has already been shown in Figure 10.

Applied inelastic fields

Thermal strains are a special case of eigenfields. When nonlinear mechanisms like plasticity are activated, the mechanical inelastic strains may appear only on certain phases and do not act as a volumetric expansion.

Consider a composite consisting of epoxy matrix and CoFe 2 O 4 fibers coated with BaTiO 3 coating layer. For this material system, two specific cases where the matrix phase is subjected to inelastic strains are discussed:

• Case 1: Inelastic normal strains ε p 11 0 " ´2ε p 22 0 " ´2ε p 33 0 " 0.001.

Figure 12 shows the macroscopic and average microscopic responses of the investigated composite under inelastic normal strains in the matrix. The evolution of the macroscopic normal stresses, axial electric displacement and axial magnetic induction with respect to the fiber volume fraction c 1 is illustrated in 12 ratio φ " 50%. It can be noticed that all macroscopic fields present significant change tendency at very high fiber volume fractions (i.e., when the matrix volume fraction is close to zero). For the microscopic fields (Figure 12 b ), similar results to those of the previous study (presence of thermal strains) are observed. Thus, electric displacement and magnetic induction parallel to the fiber axis direction are generated only in phases where piezoelectric and piezomagnetic coupling terms, respectively, are non zero.

• Case 2: shear strain p 0 " 0.001.

Figure 13 shows macroscopic and average microscopic responses of the investigated composite under inelastic axial shear strains in the matrix. The evolution of the macroscopic axial shear stress, transverse electric displacement and transverse magnetic induction with respect to the fiber volume fraction c 1 is given in Figure 13 a for ratio φ " 50%. It can be noticed that all macroscopic fields present significant change tendency at very high fiber volume fractions (i.e., when the matrix volume fraction is close to zero). For the microscopic fields (Figure 13 b ) one observes that all phases experience electric displacement and magnetic induction. This is explained by the fact that, according to the interaction tensors forms ( 43), ( 44), ( 45) and ( 46), axial shear strain can activate electric and magnetic field.

Thus, electric displacement and magnetic induction can also be generated in phases where electromechanical or magnetomechanical couplings do not exist.

Concluding comments

A micromechanical method was proposed for the evaluation of the electro-magneto-inelastic properties of coated long fiber composites with transversely isotropic piezoelectric-piezomagnetic behaviour. The method was based on solving specific boundary value problems (axial shear/in plane electric and magnetic field, transverse shear strain, plane strain/axial electric and magnetic field, hydrostatic strain/axial inelastic strain) and was adapted to the Mori-Tanaka homogenization scheme. The capabilities of the proposed micromechanics technique were verified through several as a piezomagnetic material, BaTiO 3 as a piezoelectric material and both glass and epoxy as mechanical material without magneto-or electro-mechanical coupling) for the needs of the numerical applications. A comparison with existing homogenization techniques in terms electro-magnetomechanical properties of long fiber composites was first conducted. The simulation results of the proposed approach were in good agreement with the full field homogenization results. Small difference has been observed when comparing the electromagnetic coupling terms with studies conducted by [START_REF] Dinzart | Magneto-electro-elastic coated inclusion problem and its application to magneticpiezoelectric composite materials[END_REF]. This difference was mainly attributed to the finite value of the fibers' aspect ratio. The ability of the model to predict both macroscopic and average microscopic fields per phase when nonlinear fields are activated was then demonstrated. Three cases of applied inelastic fields were considered: thermal strains, inelastic normal strains and inelastic shear strains. The effect of the inelastic fields on the overall response, as well as in the average response of the phases, was then explicitly investigated. The proposed micromechanics approach is able to handle microstructures with aligned or non-aligned fiber composites with piezoelectricpiezomagnetic behaviour under inelasticity conditions.

A. Computational steps for obtaining the interaction tensors

The unknown constants Ξ q,i and Z q,i of the boundary value problems presented in subsection 3.2 are identified using i) the boundary conditions at r " r ext Ñ 8, ii) the consistency condition that the fields should be finite at r " 0, and iii) the interface conditions ( 25) and ( 26) between the material phases.

Before presenting the solution of the four boundary value problems, the following helpful matrices and vectors are introduced: Let's consider an arbitrary material parameter ω. For the three phases parameter q for q " 0, 1, 2. The next matrices and vectors

K ω " » - - - - - - - - - - - - - - 1 ´1 ´1 0 ω 1 ´ω2 ω 2 0 0 1 φ 0 ω 2 ´ω2 φ ω 0 φ fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , K mix ω " » - - - - - - - - - - - - - - 0 0 0 0 ω 1 ´ω2 ω 2 0 0 0 0 0 0 ω 2 ´ω2 φ ω 0 φ fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl , (A.1) F ω " » - - - - - - - 0 0 1 ω 0 fi ffi ffi ffi ffi ffi ffi fl , F mix ω " » - - - - - - - 0 0 0 ω 0 fi ffi ffi ffi ffi ffi ffi fl , F 0 ω " » - - - - - - - 0 0 0 ´ω0 fi ffi ffi ffi ffi ffi ffi fl , F 1 ω " » - - - - - - - 0 ω 1 0 0 fi ffi ffi ffi ffi ffi ffi fl , F 2 ω " » - - - - - - - 0 ´ω2 0 ω 2 fi ffi ffi ffi ffi ffi ffi fl , (A.2)
have general description for the arbitrary ω and they become specific, once the ω q are assigned to proper material parameters.

A.1. Axial shear / in-plane electric and magnetic field

For this boundary value problem, the boundary conditions and the fact that all fields should be finite at r " 0 yield

Ξ 1,2 " Ξ e 1,2 " Ξ m 1,2 " 0, Ξ 0,1 " β, Ξ e 0,1 " β e , Ξ m 0,1 " β m . (A.3)
The rest of unknown constants are given by the solution of the system

K q xz •Ξ q " βF q xz `βe F q e xz `βm F q m xz `2 ÿ q"0 " s q F q p,q xz `se q F q pe,q xz `sm q F q pm,q xz ı , (A.4) where Ξ q " » - - - - - - - Ξ Ξ e Ξ m fi ffi ffi ffi ffi ffi ffi fl , F q xz " » - - - - - - - F xz F e xz F m xz fi ffi ffi ffi ffi ffi ffi fl , F q e xz " » - - - - - - - ´Fe xz F ee xz F em xz fi ffi ffi ffi ffi ffi ffi fl , F q m xz " » - - - - - - - ´Fm xz F em xz F mm xz fi ffi ffi ffi ffi ffi ffi fl , (A.5) F xz » - - - - - - - F p,q xz F pe,q xz F pm,q xz fi ffi ffi ffi ffi ffi ffi fl , F q pe,q xz " » - - - - - - - ´Fpe,q xz F pee,q xz F pem,q xz fi ffi ffi ffi ffi ffi fl , F q pm,q xz " » - - - - - - - ´Fpm,q xz F pem,q xz F pmm,q xz fi ffi ffi ffi ffi ffi ffi fl , (A.6) and K q xz " » - - - - K xz ´Ke xz ´Km xz K e xz K ee xz K em xz K m xz K em xz K mm xz fi ffi ffi ffi fl , (A.7) with Ξ " » - - - - - - - Ξ 1,1 Ξ 2,1 Ξ 2,2 Ξ 0,2 fi ffi ffi ffi ffi ffi ffi fl , Ξ e " » - - - - - - - Ξ e 1,1 Ξ e 2,1 Ξ e 2,2 Ξ e 0,2 fi ffi ffi ffi ffi ffi ffi fl , Ξ m " » - - - - - - - Ξ m 1,1 Ξ m 2,1 Ξ m 2,2 Ξ m 0,2 fi ffi ffi ffi ffi ffi ffi fl . (A.8)
The various vectors and matrices are given in Table A.1

The solution of this system can be written in the form Ξ q " βΞ q mec `βe Ξ q elc `βm Ξ q mag `2 ÿ q"0 " s q Ξ q p,q mec `se q Ξ q p,q elc `sm q Ξ q p,q mag ı , (A.9) which can be split in three parts,

Ξ " βΞ mec `βe Ξ elc `βm Ξ mag `2 ÿ q"0 " s q Ξ p,q mec `se q Ξ p,q elc `sm q Ξ p,q mag ‰ , Ξ e " βΞ e mec `βe Ξ e elc `βm Ξ e mag `2 ÿ q"0 " s q Ξ pe,q mec `se q Ξ pe,q elc `sm q Ξ pe,q mag ‰ , Ξ m " βΞ m mec `βe Ξ m elc `βm Ξ m mag `2 ÿ q"0 " s q Ξ pm,q mec `se q Ξ pm,q elc `sm q Ξ pm,q mag ‰ . (A.10)
The average strains, electric fields and magnetic fields in the fiber and the coating are given by

ε i " Λ i xz " 0 0 0 0 1 0 ı T , e i " Λ e,i xz " 1 0 0 ı T , h i " Λ m,i xz " 1 0 0 ı T , i " 1, 2, (A.11) ω µ e 15 κ e 11 κ m 11 j 11 K ω K xz K ee xz K mm xz K mix ω K e xz K m xz K em xz F ω F xz F ee xz F mm xz F mix F e xz F m xz F em xz F q ω F p,q xz F pe,q xz F pm,q xz F pee,q xz F pmm,q xz F pem,q xz Table A
.1: Axial shear / in-plane electric and magnetic field: matrices and vectors used to compute the unknown constants. Each element in this table is computed by substituting to the general matrices or vectors a specific parameter for ω. The superscript q takes the value 0, 1 or 2.

with

Λ 1 xz " Ξ 1,1 , Λ 2 xz " β `rΞ 0,2 ´Λ1 xz sφ 1 ´φ , Λ e,1 xz " Ξ e 1,1 , Λ e,2 xz " β e `rΞ e 0,2 ´Λe,1 xz sφ 1 ´φ , Λ m,1 xz " Ξ m 1,1 , Λ m,2 xz " β m `rΞ m 0,2 ´Λm,1 xz sφ 1 ´φ . (A.12)
The Λ i xz , Λ e,i xz and Λ m,i xz can be expressed in the same forms as (A.10).

A.2. Transverse shear

The boundary conditions and the fact that the displacement should be finite at r " 0 yield

Ξ 1,3 " Ξ 1,4 " Ξ 0,1 " 0, Ξ 0,2 " γ. (A.13)
The rest of unknown constants are given by the solution of the system and The solution of this system can be written in the form Ξ " γΞ mec `2 ÿ q"0 s q Ξ p,q mec . (A.17)

K xy •Ξ " γF xy `2 ÿ q"0 s q F p,q xy , (A.14) with " 1,1 Ξ 1,2 Ξ 2,1 Ξ 2,2 Ξ 2,3 Ξ 2,4 Ξ 0,3 Ξ 0,4 ı
K xy " » - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ψ 1,1 1 ´ψ2,
The average strains in the fiber and the coating are given by ε i " 2Λ The Λ i xy can be expressed in the same form as (A.17).

A.3. Plane strain / axial electric and magnetic field

The boundary conditions and the fact that the displacement should be finite at r " 0 yield Ξ 1,3 " Ξ 1,4 " Ξ 0,1 " Z 1,2 " 0, Ξ 0,2 " γ, Z 0,1 " β.

(A.20)

The rest of unknown constants are given by the solution of two systems:

1. The first is written as

K xx • Z " βF xx `βe F e xx `βm F m xx `2 ÿ q"0
" s q,β F p,q xx `se q F pe,q xx `sm q F pm,q xx ‰ , (A.21) with Z " " Z 1,1 Z 2,1 Z 2,2 Z 0,2 ı T , (A.22) F F and F p,q xx " F q ω for ω " 2K tr , (A.23)

F e xx "
2 ÿ q"0 F q ω and F pe,q xx " ´Fq ω for ω " e 31 , F m xx " 2 ÿ q"0 F q ω F pm,q xx " ´Fq ω for ω " f 31 , (A.24)

and The solution of this system can be written in the form Z " βZ mec `βe Z elc `βm Z mag `2 ÿ q"0 " s q Z p,q mec `se q Z p,q elc `sm q Z p,q mag ‰ .

K xx " » - - - - - - - - - - - - - - 1
(A.26)

2. The second is written as K xy •Ξ " γF xy `2 ÿ q"0 s q,γ F p,q xy , (A.27)

where Ξ, F xy , F p,q xy are given by (A.15) and K xy is given by (A.16).

The average strains, electric fields and magnetic fields in the fiber and the coating are given by

ε i " Λ i xx " 1 1 0 0 0 0 ı T `Λi xy " 1 ´1 0 0 0 0 ı T , e i " β e " 0 0 1 ı T , h i " β m " 0 0 1 ı T , (

A.5.

Regrouping results of the previous subsections, one finds that the interaction tensor components in (41)-( 52) are given by the expressions

Figure 2 :

 2 Figure 2: Coated cylindrical fiber with homothetic topology inside an infinite medium, which is represented as a concentric cylinder. All the fields are expressed in cylindrical coordinates.

(U

  pqq pr, θ, zq, ( E pqq pr, θ, zq, ( Σ pqq pr, θ, zq, ( E ppqq pr, θ, zq, @r, θ, z P Ω q , while the equilibrium, electrostatic and magnetostatic equations are re-expressed as B

SS

  pr 1 , θ, zq, S " rr, rθ or rz, pr 2 , θ, zq, S " rr, rθ or rz,

Figure 3 :

 3 Figure 3: Boundary value problem for axial shear / in-plane electric and magnetic field conditions.

Figure 4 :

 4 Figure 4: Boundary value problem for transverse shear strain conditions.

Figure 5 :

 5 Figure5: Boundary value plane strain / axial electric and magnetic field conditions. Case of uniaxial strain in the x 1 direction.

(

  u r ext " βr ext , ( u θ ext " 0 and ( u z ext " βz ext .

Figure 6 :

 6 Figure 6: Boundary value problem for hydrostatic strain / axial inelastic strain conditions.

Figure 7 :

 7 Figure 7: Cross section of unidirectional fiber composite. The fibers are coated and all phases exhibit piezoelectricpiezomagnetic behaviour.

Figure 8 :

 8 Figure 8: Material properties as a function of the fiber volume fraction for a composite consisting of CoFe 2 O 4 matrix and BaTiO 3 long fibers: (a) mechanical, (b) piezoelectric and (c) electromagnetic coefficients. Comparison of the developed method predictions (solid lines) with numerical results obtained by Lee et al. (2005) (points).

Figure 9 :

 9 Figure 9: Electromagnetic properties as a function of the coated fiber volume fraction of a composite consisting of CoFe 2 O 4 matrix and glass fibers coated with BaTiO 3 coating layer: Comparison of the developed method predictions (solid lines) with numerical results obtained by Dinzart and Sabar (2011) (points).

Fiber

Figure 10 :

 10 Figure 10: Composite consisting of CoFe 2 O 4 matrix and glass fibers coated with BaTiO 3 coating layer. Macroscopic stress, electric displacement and magnetic induction fields, caused by thermal strains: (a) ratio φ " 50% and (b) ratio

Figure 11 :

 11 Figure 11: Composite consisting of CoFe 2 O 4 matrix and glass fibers coated with BaTiO 3 coating layer. Average microscopic stress, electric displacement and magnetic induction fields per phase, caused by thermal strains, for fiber volume fraction 20% and ratio φ " 50%.

FiberσFigure 12 :

 12 Figure 12: Composite consisting of epoxy matrix and CoFe 2 O 4 fibers coated with BaTiO 3 coating layer, subjected to inelastic normal strains. Stress, electric displacement and magnetic induction, caused by inelastic normal strains, for ratio φ " 50%: (a) macroscopic fields and (b) average microscopic fields per phase for fiber volume fraction 20%.

FiberFigure 13 :

 13 Figure 13: Composite consisting of epoxy matrix and CoFe 2 O 4 fibers coated with BaTiO 3 coating layer, subjected to inelastic normal strains. Stress, electric displacement and magnetic induction, caused by inelastic axial shear strains, for φ " 50%: (a) macroscopic fields and (b) microscopic fields per phase for fiber volume fraction 20%.

  2 " rT xx s mec `rT xy s mec ‰ 1 2 " rT xx s mec ´rT xy s mec ‰ rT zz s mec ´rT xx s rT xx s mec ´rT xy s mec ‰ 1 2 " rT xx s mec `rT xy s mec ‰ rT zz s mec ´rT xx s

								fi
	1 2	mec " mec 0 0 1 0 0 0 0 0 0	0 0 0 rT xy s mec 0	0 0 0 0 rT xz s mec	0 0 0 0 0	ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl ffi ffi ffi ffi ffi ffi ffi ffi ffi	,
		0	0	0	0	0	rT xz s mec

  purpose of this section is to demonstrate the developed micromechanics technique capabilities. For the needs of the numerical examples, four different materials are examined: CoFe 2 O 4 , BaTiO 3 , glass and epoxy. The first one exhibits piezomagnetic behaviour, the second has piezoelectric characteristics and the last two present a typical mechanical response without magnetomechanical or electromechanical couplings. The material parameters of these materials are summarized in Table1.

		CoFe 2 O 4	BaTiO 3	glass	epoxy
	n [GPa]	269.5	162	88.8	5.53
	l [GPa]	170.5	78	29.6	2.97
	K tr [GPa]	229.5	121.5	59.2	4.25
	µ tr [GPa]	56.5	44.5	29.6	1.28
	µ ax [GPa]	45.3	43	29.6	1.28

Table 1 :

 1 Electro-magneto-mechanical properties for the materials utilized in the numerical examples. The properties of CoFe 2 O 4 , BaTiO 3 and epoxy have been obtained from

  K 31 K 32 K 33 K 34 K 35 K 36 0 0 K 41 K 42 K 43 K 44 K 45 K

	K 73 r2K	r2ψ 2,1 ´1s `2µ tr 2 r1 `ψ2,1 ss {φ, K 74 " 2µ tr 2 ,
	K 75 " 6µ tr 2 φ 2 , K 76 " ´r2µ tr 2 rψ 2,4 ´1s `2K tr 2 s φ,
	K 77 " tr 0 φ 2 , K 78 " r2µ tr 0 rψ 0,4 ´1s `2K tr 0 s φ,
	K 83 " 2µ tr 2 r1 `ψ2,1 s{φ, K 84 " 2µ tr 2 ,
	K 85 " ´6µ tr 2 φ 2 , K 86 " ´2µ tr 2 r1 ´ψ2,4 sφ,
	K 87 " 6µ tr 0 φ 2 , K 88 " 2µ tr 0 r1 ´ψ0,4 sφ.
	1 0 0 0	1 ´1 1 ´ψ2,4 0 ´1 ´1 ´1 0 46 0 0 ψ 2,1 {φ 1 ´φ2 ψ 2,4 φ φ 2 1 ´1 0 1{φ φ 2 φ ´φ2 0 K 73 K 74 K 75 K 76 K 77	0 0 0 ´ψ0,4 φ ´φ K 78	fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi	,	(A.16)
	0	0	K 83 K 84 K 85 K 86 K 87	K 88
	with
	K 31 " 2K tr 1 r2ψ 1,1 ´1s `2µ tr 1 r1 `ψ1,1 s, K 32 " 2µ tr 1 ,
	K 33 " ´2K tr 2 r2ψ 2,1 ´1s ´2µ tr 2 r1 `ψ2,1 s, K 34 " ´2µ tr 2 ,
	K 35 " ´6µ tr 2 , K 36 " 2µ tr 2 rψ 2,4 ´1s `2K tr 2 ,
	K 41 " 2µ tr 1 r1 `ψ1,1 s, K 42 " 2µ tr 1 ,
	K 43 " ´2µ tr 2 r1 `ψ2,1 s, K 44 " ´2µ tr 2 ,
	K 45 " 6µ tr 2 , K 46 " 2µ tr 2 r1 ´ψ2,4 s, 39

  xy are the same with those of the expressions (A.19). The Λ i xx can be expressed in the same form as (A.26).

					A.28)
	with
	and Λ 1 xy , Λ 2	Λ 1 xx " Z 1,1 , Λ 2 xx "	β `rZ 0,2 1 ´φ ´Λ1 xx sφ	,	(A.29)

In classical magnetomechanics studies, the natural choice is to introduce a vector potential. When dealing with multiscale approaches, the absence of microscopic current density permits an alternative formalism with scalar potential, which simplifies the calculations.

The symbol • in the matrix notation denotes the common matrix multiplication. The product between a scalar and a matrix is represented without a symbol.

A.4. Hydrostatic

For value problem, the boundary conditions and the fact that the displacement should be finite at r " 0 yield Z 1,2 " 0, Z 0,1 " β.

(A.30)

The rest of unknown constants are given by the solution of the system

and K xx is the same with (A.25). The solution of this system can be written in the form Z " βZ mec `2 ÿ q"0 s r Z p,q mec . (A.34)

The average strains in the fiber and the coating are given by

The Λ i zz can be expressed in the same form as (A.34).