Effective properties of periodic fibrous electro-elastic composites with mechanic imperfect contact condition

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Effective properties of periodic fibrous electro-elastic composites with mechanic imperfect contact condition

Reinaldo Rodrıguez-Ramos, Raul Guinovart-Dıaz, Juan Carlos Lopez-Realpozo, Julian Bravo-Castillero, José Otero, Frederico Sabina,

Frédéric Lebon

To cite this version:

Reinaldo Rodrıguez-Ramos, Raul Guinovart-Dıaz, Juan Carlos Lopez-Realpozo, Julian Bravo-

Castillero, José Otero, et al.. Effective properties of periodic fibrous electro-elastic composites with

mechanic imperfect contact condition. International Journal of Mechanical Sciences, Elsevier, 2013,

73, pp.1-13. �10.1016/j.ijmecsci.2013.03.011�. �hal-00861434�

Effective properties of periodic fibrous electro-elastic composites with mechanic imperfect contact condition

In this work, two-phase parallel fiber-reinforced periodic piezoelectric composites are considered wherein the constituents exhibit transverse isotropy and the cells have different con fi gurations.

Mechanical imperfect contact at the interface of the composites is studied via linear spring model. The statement of the problem for two phase piezoelectric composites with mechanical imperfect contact is given. The local problems are formulated by means of the asymptotic homogenization method (AHM) and their solutions are found using complex variable theory. Analytical formulae are obtained for the effective properties of the composites with spring imperfect type of contact and different parallelogram cells. Some numerical examples and comparisons with other theoretical results illustrate that the model is efficient for the analysis of composites with presence of parallelogram cells and the aforementioned imperfect contact.

1. Introduction

Nowadays, piezoelectric materials have a key role in manufac- turing of sensors and actuators, which may be used for active control of elastic deformations and vibrations of the structures.

These materials have a wide range of applications in science and technology such as in ultrasonic transducers, sonar projects, and under water acoustic.

In order to successfully integrate piezoelectric actuators into structures, the physical nature of the interface condition between the actuators and the base structure, and its effect on the induced electro-mechanical fi eld must be fully understood.

Some years ago, Hashin [1] using the generalized self-consistent scheme studied the thermoelastic properties of unidirectional fi ber composites with imperfect interface conditions de fi ned in terms of linear relations between interface tractions and displacement jumps. Besides, the asymptotic scheme for the analysis of dilute elastic composites, which includes circular inclusions with imper- fect bonding at the interface, is presented by Bigoni et al. [2]. The interface, in this work, is characterized by a discontinuous displace- ment fi eld across it, linearly related to the tractions. Recently, an

asymptotic approach for simulation of the imperfect interfacial bonding in composite materials is proposed by Andrianov et al.

[3] where a problem of the axial shear of elastic fi bre-reinforced composites with square and hexagonal arrays of cylindrical inclu- sions is considered. The performed analysis is based on the asymptotic homogenization method and the cell problem is solved using the underlying principles of the boundary shape perturbation technique. Moreover, the effective elastic moduli of composite materials are investigated by Yanase and Ju [4] in the presence of imperfect interfaces between the inclusions and the matrix. The primary focus is on the spherical particle reinforced composites. By admitting the displacement jumps at the particle – matrix interface, the modi fi ed Eshelby inclusion problem is studied. Besides, Chen et al. [5] studied a micromechanical method based on generalized method of cells for investigating elastic and plastic response of composites subjected to off-axis loading. To date, to the best of the author ’ s knowledge, the problems associated with piezoelectric materials and inhomogeneities with imperfect interface conditions have not been reported intensively in the literature. For instance, piezothermoelastic constitutive laws at a weak interface is analyzed by Shu [6]; Shodja et al. [7] examine the electro-mechanical fi elds for a circular anisotropic piezoelectric fi ber sensor inside an anisotropic piezoelectric or non-piezoelectric elastic matrix with imperfect interface under remote in-plane uniform tension, among other works.

n

Corresponding author. Tel.:

þ53 7 832 2466.

E-mail address:

[email protected] (R. Rodríguez-Ramos).

a

Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L Vedado Habana 4, CP 10400, Cuba

b

Instituto de Cibernética, Matemática y Física, ICIMAF. Calle 15 No. 551, entre C y D. Vedado, Habana 4, CP 10400, Cuba

c

Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Universidad Nacional Autónoma de México, Apartado Postal 20-726 Delegación de Álvaro Obregón, 01000 México, DF, México

d

Laboratoire de Mécanique et d’Acoustique, Université Aix-Marseille , CNRS, Centrale Marseille, 31 Chemin Joseph-Aiguier, 13402 Marseille Cedex 20, France

R. Rodríguez-Ramos

^a,*

, R. Guinovart-Díaz

^a

, J.C López-Realpozo

^a

, J. Bravo-Castillero

^a

,J.A. Otero

^b

, F.J. Sabina

^c

, F. Lebon

^d

Different authors are investigating the behavior of composites with non-perfect bonding contact. Recently, an asymptotic study of different types of imperfect interfaces arising in the problem of conduction through a granular composite material was presented in [8]. In Andrianov et al. [3], imperfection is considered by means of a discontinuity of the displacement (spring model). However, it may look natural that the mechanical weakening of the interface (due to delamination, decohesion, etc.) should also induce the decrease of the electric contact. For instance, in piezoelectric material, due to their electro-mechanical coupling, there exist induced electric charges when a mechanical loading is applied [9].

Although the electro-mechanical coupling exists, the mechanical contribution is remarkable in the behavior of the composites. In this sense, as a fi rst approximation in the study of piezoelectric composites under imperfect contact (mechanic and electric) we assume only mechanical imperfect adherence.

The present work is motivated by the interest to study the in fl uence of imperfect contact over the effective piezoelectric response when the composites have oblique fi brous orientation.

Composites with rhombus periodic cell are important since they could describe monoclinic behavior of certain physical and biolo- gical structures. This is an extension of previous results reported by Bravo-Castillero et al. [10] and Sabina et al. [11] where perfect contact for piezoelectric composites was considered. Moreover, in this contribution other recent researches related to composites with perfect contact conditions and parallelogram cells studied by Guinovart-Díaz et al. [12,13] and Rodríguez-Ramos et al. [14] are extended to composites with the same distribution of the periodic cells but now with no-well bonding contact. The interface imper- fection is posed on the mechanical fi elds only. The mechanical behavior of imperfect interface is modeled via an idealization of a layer of mechanical springs of zero thickness. The vanishing value of K ~

n

and K ~

t

, K ~

s

corresponds to pure debonding (normal perfect debonding), in-plane pure sliding, and out-of-plane pure sliding, respectively. The status of the mechanical bonding is completely determined by appropriate values of these constants. For large enough values of the constants, the perfect bonding interface is achieved. The spring approach is used for the calculation of the piezoelectric effective coef fi cients in a composite with different angular distribution of fi bers. Using the two scale asymptotic homogenization method the formulation of the local problems for linear two phase piezoelectric composites with parallelogram cell and mechanical imperfect contact conditions is given and the solution of each plane local problems is found using the potential methods of a complex variable and the properties of doubly periodic Weierstrass elliptic functions. Besides, the complete set of analytical expressions for the piezoelectric coef fi cients of a fi ber reinforced composite with circular cylindrical shape periodically distributed in the matrix under linear spring imperfect contact conditions are obtained via AHM. The study of such composites with mechanic and electric coupled effect is an extension of previous works considered by Molkov and Pobedria [15], Rodríguez-Ramos et al. [16] and Lopez-Realpozo et al. [17] where only the elastic properties of the composite with mechanical imperfect contact were analyzed. In particular, the last two works are referred only to antiplane elastic properties.

The heterogeneous problem formulation is presented in Section 1 where the basic equations and the general statement of the imperfect conditions are written. In Section 2 the two scale asymptotic homogenization algorithm is developed and the state- ment of the plane and antiplane local problems with mechanical (spring) imperfect conditions are written. Solutions of each local problem are given in Sections 3 and 4. Moreover, Section 5 is devoted to present some important parameters used for evaluat- ing the performance of 1 – 3 piezoelectric composites. In Section 6 validations of the present model and comparisons with other

theoretical and experimental approaches are shown as well as the effect of the imperfect adherence in the ultrasonic transducers applications. Finally, some conclusions are written.

2. Heterogeneous problem formulation

Consider piezoelectric materials that respond linearly to changes in the mechanic and electric fi elds. A two-phase uniaxial reinforced material is considered here in which fi bers and matrix have homo- geneous and transversely isotropic properties; the axis of transverse symmetry coincides with the fi ber direction, which is taken as the Ox

3

-axis. The fi ber cross-section is circular. Moreover, the fi bers are periodically distributed without overlapping in directions parallel to the Ow

1

- and Ow

2

-axis, where w

1

≠ 0 and w

2

≠ 0 ðw

2

≠λ w

1

, λ∈ℝ Þ are two complex numbers which de fi ne the parallelogram periodic cell of the two-phase composite. Therefore the composite Ω consists of a parallelogram array of identical circular cylinders embedded in a homogeneous medium (Fig. 1). The cylinders are infinitely long.

The response of the material at the microscale level is analyzed using representative material elements (RME) or Representative Cell.

The fi ber-reinforced material is assumed to have a periodic arrange- ment of fi ne scale fi bers embedded in a matrix. A sample RME is shown in Fig. 2 where the appropriate periodic unit cell Y is taken as a regular parallelogram in the y

1

y

2

-plane so that Y ¼ Y

1

∪ Y

2

with Y

1

∩ Y

2

¼ ϕ , the domain Y

1

is occupied by the matrix and its complement Y

2

a circle of radius R, is fi lled up with the fi ber for a piezoelectric composite with rectangle, rhombic and parallelogram arrangements of unidirectional fi bers. A local Cartesian coordinate system y is introduced at the microscale and oriented such that the y

₃

-axis is aligned parallel to the axis of the fi bers. The microscale coordinates y of a point in the RME are related to the macroscale coordinates x by y ¼ x =ε , where ε≪ 1. Beside the use of subscript, matrix and fi ber associated quantities are also referred below by means of superscripts in brackets (1) and (2), respectively. Two- phase composite is considered which comprises a matrix with homogeneous properties given by the following moduli tensors:

elastic C

^ð1Þ_ijkl

, piezoelectric e

^ð1Þ_ijk

and dielectric permittivity κ

^ð1Þij

, in which are embedded parallel circular cylindrical fi bers with corresponding homogeneous properties C

^ð2Þ_ijkl

, e

^ð2Þ_ijk

and κ

^ð2Þij

:

To denote the dependence of a fi eld variable on the macroscale and microscale coordinates, the superscript ε is used, i.e.

ℱ

^ε

¼ ℱ ðx,yÞ where ℱ

^ε

represents a scalar, vector or tensor fi eld.

All fi eld variables are assumed to depend on the coordinates of both scales. For an arbitrary microstructure, material phases, and therefore material properties such as elastic constants, are

Fig. 1.

The heterogeneous medium and extracted the parallelogram periodic cell.

functions of the microscale coordinates y : In the following for- mulation, indicial and direct notation will be used interchangeably.

The local governing mechanical and charge equilibrium equa- tions in the absence of body forces and free charges are

s

^εij,j

¼ 0, D

^ε_i,i

¼ 0 in Ω ð1Þ

where the subscript comma denotes partial differentiation, s

^εij

are the components of the Cauchy stress tensor and D

^ε_i

are the components of the electric displacement vector.

Using the conventional indicial notation in which repeated subscripts are summed over the range of i,j,k,l ¼ 1,2,3, the con- stitutive equations are

s

^εij

¼ C

_ijkl

ε

^εkl

− e

_kij

E

^ε_k

, D

^ε_i

¼ e

_ikl

ε

^εkl

þ κ

ik

E

^ε_k

, ð2Þ where ε

^εkl

is the in fi nitesimal strain tensor and E

^ε_k

, is the electric fi eld vector. The quantities C

ijkl

, e

kij

, κ

ik

are components of the elastic stiffness tensor, the piezoelectric tensor, and the dielectric permittivity tensor, respectively.

The gradient equations, which involve the strain – displacement equations and electric fi eld-potential, are

ε

^εkl

¼ 1 2

∂ u

^ε_k

∂ x

l

þ ∂ u

^ε_l

∂ x

k

, E

^ε_k

¼ −ϕ

;^εk

, ð3Þ

where u

^ε_k

and ϕ

^ε

are the mechanical displacement and the electric potential, respectively.

The material constants are assumed to satisfy the symmetries C

ijkl

¼ C

jikl

¼ C

klij

, e

kij

¼ e

kji

, κ

ik

¼ κ

ki

: In addition, the elasticity ten- sor and the dielectric permittivity tensor are assumed to be positive de fi nite.

Substituting (2) and (3) into (1) we obtain a coupled system of partial differential equations with coef fi cients rapidly oscillating ðC

ijkl

ðyÞu

^ε_k,l

þe

kij

ðyÞ ϕ

^ε,k

Þ

_,j

¼ 0, ðe

ikl

ðyÞu

^ε_k,l

−κ

ik

ðyÞ ϕ

^ε,k

Þ

_,i

¼ 0 in Ω : ð4Þ

Eq. (4) represents a system of equations for fi nding u

i

and ϕ . For a complete solution, it is necessary to assign suitable boundary conditions, for instance

u

^ε_i

¼ u ^

i

; s

^εij

n

j

¼ S ^

i

; ϕ

^ε

¼ ϕ

0

; D

^ε_i

n

i

¼ 0 on ∂Ω , ð5Þ where u ^

i

, S ^

_i

and ϕ

0

are the prescribed displacement, force and electric potential on the boundary of the composite, respectively.

The interface conditions are speci fi ed as follows. The inclusion problems associated with piezocomposite materials, which have been presented in the literature, are mainly concerned with perfect interface condition; see for example, the works of Avella- neda and Swart [18] and Gibiansky and Torquato [19]. In the case of perfect bonding, the continuities of displacement, traction, electric potential, and normal electric displacement are concerned.

Often, the above electro-mechanical interface conditions are not realistic assumptions in modeling the actual physical problems. In this section it is intended to analyze the behavior of a piezo- composite under only mechanical imperfect contact. The mechan- ical behavior of imperfect interface is modeled via an idealization of a layer of mechanical springs of zero thickness. The spring

constants are the measures for the magnitude of the associated continuities. The vanishing value of K ~

n

and K ~

t

, K ~

s

corresponds to pure debonding (normal perfect debonding), in-plane pure sliding, and out-of-plane pure sliding, respectively. The status of the mechanical bonding is completely determined by appropriate values of these constants. For large enough values of the constants, the perfect bonding interface is achieved.

The spring stiffness matrix, the mechanic displacement and the traction vectors using the vector notation are written as

u ¼ u

n

u

t

u

s

0 B @

1 C A , T ¼

T

n

T

t

T

s

0 B @

1 C A , K ¼

K ~

n

0 0 0 K ~

t

0 0 0 K ~

s

0 B @

1

C A: ð6Þ

The effect of mechanical imperfection is incorporated through the mechanical displacements jumps across the interface, while the corresponding tractions, electric potential and normal electric displacement remain continuous. Several examples addressing the effect of electro-mechanical imperfections on the induced electro- mechanical fi elds are thoroughly examined by Shodja et al. [9].

There are seven types of imperfections considered in Table 1 of the work of Shodja et al. [9]. Mechanical partial debonding (type of imperfection VII) listed in Table 1 is focused in the present work.

Hence, the mechanical imperfect condition considered in Shodja et al. [7] and Hashin [1] may be expressed as

T

^ð1Þ

þT

^ð2Þ

¼ 0, T

^ðγÞ

¼ ð − 1Þ

^γþ1

K½u, ½ ϕ ¼ 0, ½Dn ¼ 0 on Γ: ð7Þ In these relations ½ indicates the jump in the quantity at the common interface Γ between the fi ber and the matrix; n is the outward unit normal on Γ ; u

n

, u

t

, u

s

are the normal and the two tangential components of the mechanic displacement vector, respectively; T

n

, T

t

, T

s

are the normal and tangential components of the traction vector T (T

i

¼ s

ij

n

j

). The superscripts ð γ Þ, γ ¼ 1,2 denote the matrix and fi ber respectively.

In order to study the imperfect contact conditions, the relations between the displacement and traction vectors (6) are related to their Cartesian representations by the following expressions:

u

n

u

t

u

s

0 B @

1 C A ¼

cosφ sinφ 0

−sinφ cosφ 0

0 0 1

0 B @

1 C A

u

1

u

2

u

3

0 B @

1 C A,

T

n

T

t

T

s

0 B @

1 C A ¼

cosφ sinφ 0

−sinφ cosφ 0

0 0 1

0 B @

1 C A

T

1

T

2

T

3

0 B @

1 C A:

ð8Þ Thus, the expression (7) on Γ , can be rewritten in the following indicial form:

T

^ð1Þ

þT

^ð2Þ

¼ 0 on Γ , T

^ð_n^γÞ

¼ ð − 1Þ

^γþ1

K ~

n

½u

n

, T

^ð_t^γÞ

¼ ð − 1Þ

^γþ1

K ~

t

½u

t

, T

^ð_s^γÞ

¼ ð − 1Þ

^γþ1

K ~

s

½u

s

on Γ , ½ ϕ ¼ 0, ½Dn ¼ 0 on Γ: ð9Þ

3. Two scales asymptotic formulation

The overall properties of the above periodic medium are sought by means of the AHM in the same way as it is developed by Bakhvalov and Panasenko [20] and Bensoussan et al. [21]. In this R

R

R y

₁

y

₂

y

₁

y

₁

y

₂

y

₂

Y

₁

Y

₂

Y

₁

Y

₁

Y

₂

Y 2 w

₁

w

2

w

₂

w

₁

w

₁

Θ

Θ Θ

w

₂

Fig. 2.

Different unit cells

–

rectangle, rhombic and parallelogram.

section, the application of the general AHM method using the two scale formulation for the electroelastic heterogeneous problem is presented. As discussed earlier, fi eld variables are assumed to depend on both the macroscale coordinates x and microscale coordinates y, and therefore, spatial derivatives of the fi eld variables are obtained by use of the chain rule and the relation y ¼ x =ε as

∂ℱ

^ε

∂ x

i

¼ ∂ℱ ðx,yÞ

∂ x

i

þ 1 ε

∂ℱ ðx,yÞ

∂ y

i

: ð10Þ

The AHM method assumes a two-scale asymptotic expansion for the mechanical displacement and electric potential analogous to Bravo-Castillero et al. [10] and Sabina et al. [11]:

u

^ε

ðxÞ ¼ u

⁰

ðx,yÞþ ε u

¹

ðx,yÞþ ⋯ ,

ϕ

^ε

ðxÞ ¼ ϕ

⁰

ðx,yÞþ εϕ

¹

ðx,yÞþ ⋯ , ð11Þ where the superscripts on the fi eld variables denote the different orders in the asymptotic expansion and do not imply exponentia- tion. Although u

^ε

and ϕ

^ε

depend on coordinates x and y, it has been previously demonstrated that the lowest order terms u

⁰

and ϕ

⁰

depend only on the macroscale coordinates x and correspond to the average macroscale values as ε- 0 : In other words,

lim

ε-0

u

^ε

¼ u

⁰

ðx,yÞ ¼ uðxÞ, lim

ε-0

ϕ

^ε

¼ ϕ

⁰

ðx,yÞ ¼ ϕ ðxÞ,

where uðxÞ and ϕ ðxÞ denote the average macroscale displacement and electric potential, respectively. The average is considered in the sense that any fi eld variable ℱ

^ε

, which is a function of both the macroscale and microscale coordinates, is integrated over the cell domain to obtain the corresponding averaged fi eld variable ℱ which is dependent only on the macroscale coordinates, i.e.

ℱ ¼ ð1 = jYjÞ R

Y

ℱ

^ε

dY :

As such, the fi rst order terms ε u

¹

ðx,yÞ and εϕ

¹

ðx,yÞ represent the microscale fl uctuations in the mechanical displacement and electric potential, respectively. The in fi nitesimal strains and elec- tric fi eld, which are obtained from (3) and (11) using (10), are substituted into the constitutive equation (2) to obtain the following expansions for the stress and electric displacement:

s

^εij

¼ s

⁰ij

ðx,yÞþOð ε Þ, D

^ε_i

¼ D

⁰_i

ðx,yÞþOð ε Þ, ð12Þ where the zero order terms for the stress and electric displace- ment are

s

⁰ij

ðx,yÞ ¼ C

ijkl

∂ u

k

∂ x

l

þ ∂ u

¹_k

∂ y

_l

!

þe

kij

∂ϕ

∂ x

k

þ ∂ϕ

¹

∂ y

_k

! ,

D

⁰_i

ðx,yÞ ¼ e

ikl

∂ u

k

∂ x

_l

þ ∂ u

¹_k

∂ y

_l

!

−κ

ik

∂ϕ

∂ x

_k

þ ∂ϕ

¹

∂ y

_k

!

: ð13Þ

Since ε≪ 1, the microscale stress and electric displacement correspond to the fi rst order term in (12). Inserting (12) into the mechanical and charge equilibrium equation (1), multiplying by ε , and taking the limit as ε- 0 yields the following local equilibrium equations:

∂s

⁰ij

∂ y

j

¼ 0, ∂ D

⁰_i

∂ y

i

¼ 0 : ð14Þ

To proceed with the asymptotic formulation, due to the linearity of this problem and assuming regularity of both the inclusions shapes and the smoothness of the coef fi cients, the following decompositions are used for u

¹

and ϕ

¹

:

u

¹

ðx,yÞ ¼

_pq

MðyÞ ∂ u

⁰_p

∂ x

q

ðxÞþ

_p

PðyÞ ∂ϕ

⁰

∂ x

q

ðxÞ, ϕ

¹

ðx,yÞ ¼

_pq

NðyÞ ∂ u

⁰_p

∂ x

q

ðxÞþ

_p

Q ðyÞ ∂ϕ

⁰

∂ x

q

ðxÞ, ð15Þ

where the sets of pq-functions,

_pq

Μ ðyÞ and

_pq

NðyÞ, and p-functions,

p

PðyÞ and

_p

Q ðyÞ, depend only on y : They are microscale character- istic functions that relate the macroscale strain and electric fi eld to the microscale fl uctuations in the mechanical displacement and electric potential. They are the unique periodic solution of the so- called

_pq

-local and

_p

-local problems, denoted by

_pq

L and

_p

I, respectively, over the periodic unit cell Y, de fi ned below.

The

_pq

L problem seeks displacements

_pq

M

_ðγÞ

ðyÞ and potential

pq

N

_ðγÞ

ðyÞ, in Y

_γ

, γ ¼ 1,2, which are periodic functions of periods w

1

¼ 1, w

2

¼ be

^iΘ

, b 4 0 is the modulus of this complex number and are the solution of the following equations

pq

s

^ð_iδ,δ^γÞ

¼ 0 in Y

_γ

,

pq

D

^ð_δ,δ^γÞ

¼ 0 in Y

_γ

, ð16Þ

under imperfect mechanical and perfect electrical conditions

pq

T

^ð1Þ

þ

_pq

T

^ð2Þ

¼ 0, ½

_pq

N ¼ 0 on Γ ,

_pq

T

^ð_n^γÞ

¼ ð − 1Þ

^γþ1

K ~

n

½

_pq

M

_n

,

pq

T

^ð_t^γÞ

¼ ð − 1Þ

^γþ1

K ~

t

½

_pq

M

_t

,

_pq

T

^ð_s^γÞ

¼ ð − 1Þ

^γþ1

K ~

s

½

_pq

M

_s

on Γ ,

½

_pq

D

_δ

n

_δ

¼ − ½e

^ð_δ^γ_pq^Þ

n

_δ

on Γ: ð17Þ To assure the solution of the

_pq

L problems is unique, the functions also satisfy condition that 〈

pq

M 〉 ¼ 〈

pq

N 〉 ¼ 0, where the angular brackets de fi ne the volume average per unit length over the unit periodic cell 〈 F 〉 ¼ 1 = jYj R

Y

FðyÞdy : Moreover,

pq

s

^ði^γδ^Þ

¼ C

^ð_i^γ_δ^Þ_{kλ pq}

M

^ð_k,λ^γÞ

þe

^ð_λ^γ_i^Þ_{δ pq}

N

^ð_,λ^γÞ

,

pq

D

^ð_δ^γÞ

¼ e

^ð_δ^γ_k^Þ_{λ pq}

M

^ð_k,λ^γÞ

−κ

^ð_δλ^γÞpq

N

^ð_,λ^γÞ

, ð18Þ the comma notation denotes a partial derivative relative to the y

_δ

component, i.e. U

_,δ

≡∂ U =∂ y

_δ

; the summation convention is also understood for Greek indices, which run from 1 to 2; no summa- tion is carried out over upper case indices, whether Latin on Greek.

The functions

_pq

M

_t

,

_pq

M

_s

,

_pq

M

_n

are the two tangential and normal components of the vector

_pq

M whereas

_pq

T

_t

,

_pq

T

_s

,

_pq

T

_n

, are the two tangential and normal components of the traction vector

pq

T

_i

¼ ð

_pq

s

ij

þC

ijpq

Þn

j

associated to the local problem

_pq

L. The symmetry between the indices p and q shows right away that at most six problems needs to be considered.

In similar manner the

_p

I problem is stated as follows: the displacements

_p

P

ðγÞ

ðyÞ and potential

_p

Q

ðγÞ

ðyÞ are sought in Y

_γ

, γ ¼ 1,2, which are periodic functions of periods w

1

¼ 1, w

2

¼ b e

^iΘ

, b is the modulus of this complex number and that satisfy following equations:

p

s

^ðiδ,δ^γÞ

¼ 0 in Y

_γ

,

p

D

^ð_δ,δ^γÞ

¼ 0 in Y

_γ

, ð19Þ

and the above mentioned interface conditions adapted to this problem

p

T

^ð1Þ

þ

_p

T

^ð2Þ

¼ 0, ½

_p

Q ¼ 0, on Γ ,

_p

T

^ð_n^γÞ

¼ ð − 1Þ

^γþ1

K ~

n

½

_p

P

_n

,

p

T

^ð_t^γÞ¼ ð−

1Þ

^γþ1K~t½_pP_t, _pT^ðγÞ_s ¼ ð−

1Þ

^γþ1K~s½_pP_s,

on

Γ, ½_pD_δn_δ¼ ½κ^ðγÞδpn_δ,

on

Γ,

ð20Þ where the functions

_p

P

t

,

_p

P

s

,

_p

P

n

are the two tangential and normal components of the vector

p

P whereas

p

T

t

,

p

T

s

,

p

T

n

, are the tangential and normal components of the traction vector

p

T

i

¼ ð

p

s

^ij

þe

pij

Þn

j

associated to the local problem

_p

I and

〈

p

P 〉 ¼ 〈

p

Q 〉 ¼ 0,for the uniqueness of the solution. Furthermore,

p

s

^ðiδ^γÞ

¼ C

^ð_iδ^γÞ_{kλ p}

P

^ð_k,λ^γÞ

þe

^ð_λ^γ_iδ^Þ _p

Q

^ð_,λ^γÞ

,

p

D

^ð_δ^γÞ

¼ e

^ð_δ^γ_k^Þ_{λ p}

P

^ð_k,λ^γÞ

−κ

^ð_δλ^γÞ p

Q

^ð_,λ^γÞ

: ð21Þ Also the non-homogeneous

_p

I problems will cooperate towards the homogenized moduli.

The local equilibrium equation (14) are multiplied by test

functions v

i

ðyÞ and wðyÞ, respectively, and integrated over the

RME domain Y to obtain the weak forms of the mechanical and charge equilibrium equations:

Z

Y

s

⁰ij

∂ v

i

∂ y

_j

dY ¼ 0, Z

Y

D

⁰_i

∂ w

∂ y

_i

dY ¼ 0 : ð22Þ

Inserting (15) into (13) and the result into (22), and requiring that the equation hold for arbitrary ∂ u

⁰_p

=∂ x

q

and ∂ϕ

⁰

=∂ x

q

results in the following weak form equations for the characteristic functions, the sets of pq-functions

_pq

Μ ðyÞ and

_pq

NðyÞ, and p-functions,

_p

PðyÞ and

_p

Q ðyÞ,

Z

Y

C

ijkl

∂

pq

M

k

∂ y

_l

− e

kij

∂

pq

N

∂ y

_k

∂ v

_i

∂ y

_j

dY ¼ Z

Y

C

ijpq

∂ v

_i

∂ y

_j

dY, Z

Y

e

_kij

∂

p

Q

∂ y

k

− C

_ijkl

∂

p

P

k

∂ y

l

∂ v

i

∂ y

j

dY ¼ Z

Y

e

qij

∂ v

i

∂ y

j

dY, Z

Y

e

ikl

∂

pq

M

_k

∂ y

_l

þ κ

ik

∂

pq

N

∂ y

_k

∂ w

∂ y

_i

dY ¼ Z

Y

e

ipq

∂ w

∂ y

_i

dY, Z

Y

κ

ik

∂

p

Q

∂ y

_k

þe

ikl

∂

p

P

_k

∂ y

_l

∂ w

∂ y

_i

dY ¼ Z

Y

κ

iq

∂ w

∂ y

_i

dY : ð23Þ Eq. (23) represents a system of partial differential equations that must be solved to obtain the microscale characteristic func- tions

_pq

Μ ðyÞ,

_pq

NðyÞ,

_p

PðyÞ and

_p

Q ðyÞ. The characteristic functions are subject to periodic boundary conditions over the RME domain Y in analogous form to Pobedria [22].

The constitutive relations of the linear piezoelectric theory for a heterogeneous and periodic medium, Ω , is characterized by the Y-periodic functions CðyÞ, eðyÞ, κ ðyÞ. The original constitutive relations with rapidly oscillating material coef fi cients are trans- formed in new physical relations with constant coef fi cients C

ⁿ

, e

ⁿ

, κ

ⁿ

which represent the elastic, piezoelectric and permittivity properties, respectively of an equivalent homogeneous medium and are called the effective coef fi cients of Ω . Therefore, the system (4) can be transformed into equivalent system with constant coef fi cients which represent the overall properties of the composite.

The main problem to obtain such average formulae is to fi nd the Y-periodic solutions of nine

_pq

L,

_p

I (p,q ¼ 1,2,3) local problems on Y in terms of the fast variable y as it was reported by Bravo- Castillero et al. [10] and Sabina et al. [11] based on the mathema- tical statement of both problems.

Once the local problems are solved, the homogenized moduli C

ⁿ_ijpq

, e

ⁿ_kij

, κ

ⁿik

may be determined by using the following formulae:

C

ⁿ_ijpq

¼ 〈C

ijpq

þC

_{ijkl pq}

M

k,l

þe

_{kij pq}

N

,k

〉 , e

ⁿ_ipq

¼ 〈e

ipq

þe

_{ikl pq}

M

k,l

−κ

ik pq

N

,k

〉 , e

ⁿ_pij

¼ 〈 e

_pij

þC

_{ijkl p}

P

_k,l

þe

_{kij p}

Q

_,k

〉 , κ

ⁿ_ip

¼ 〈κ

ip

− e

_{ikl p}

P

_k,l

þ κ

ik p

Q

_,k

〉:

ð24Þ Finally, the homogenized boundary value problem associated with (4)-(5) has the form

s

⁰ij,j

¼ 0, D

⁰_i,i

¼ 0 in Ω , u

⁰_i

¼ u

_i

; s

⁰_ij

n

_j

¼ S

_i

; f

⁰

¼ ϕ

0

; D

⁰_i

n

_i

¼ 0 on ∂Ω , ð25Þ where the corresponding macroscale constitutive equations are s

⁰ij

¼ C

ⁿ_ijkl

u

⁰_k,l

þe

ⁿ_kij

ϕ

⁰,k

, D

⁰_i

¼ e

ⁿ_kij

u

⁰_k,l

−κ

ⁿik

ϕ

⁰,k

: ð26Þ

Here u

⁰_i

ðxÞ ¼ 〈 u

i

ðx,yÞ 〉 is the averaged displacement vector and ϕ

⁰

ðxÞ ¼ 〈ϕ ðx,yÞ 〉 the averaged electric potential.

Each local problem (16) and (17) and (19) and (20) ðp,q ¼ 1,2,3Þ uncouples into two sets of equations. The plane and antiplane- strain systems of equations which correspond to fi ve plane-strain local problems

_pp

L,

₁₂

L,

₃

I and fourth antiplane-strain one

13

L,

₂₃

L,

₁

I,

₂

I : Table 1 shows the correspondence between the effective properties and the local problems. The global behavior of the piezoelectric composite is related to the class symmetry monoclinic 2, see details in Royer and Dieulesaint [23], which

contain 13 elastic, 8 piezoelectric and 4 dielectric independent coef fi cients.

The local-value problems set up in (16) and (17), (19) and (20) have been solved in the present work using the methods of a complex variable and the properties of doubly periodic elliptic and related functions with periods w

1

and w

2

as it is reported by Bravo-Castillero et al. [10] and Sabina et al. [11]. Taking into account that the rate of debonding would depend not only upon the debonding parameters, but also upon the elastic moduli of the components and the fi ber volume fraction [1] in the solution of these problems, the following relations are used K ~

t

¼ K

t

C

^ð1Þ₄₄

= R, K ~

n

¼ K

n

C

^ð1Þ₄₄

= R, K ~

s

¼ K

s

C

^ð1Þ₄₄

= R where K

t

,K

n

and K

s

are dimensionless parameters.

4. Solution of antiplane problems

Now, the problem

₁₃

L is explained in detail from the set of antiplane problems

₁₃

L,

₂₃

L,

₁

I,

₂

I. From now on, the preindices are not used and the effective properties are denoted with the short notation. The determination of the shear piezoelectric effective properties, denoted by C

ⁿ₄₄

, C

ⁿ₄₅

, C

ⁿ₅₅

, (shear moduli), e

ⁿ₁₅

, e

ⁿ₁₄

, e

ⁿ₂₄

, (shear stress piezoelectric coef fi cient) and κ

ⁿ11

, κ

ⁿ12

, κ

ⁿ22

, (transverse permittivity constant) is the main aim of this part where the constituents of each phase of the composite are of class 6 mm and the short indicial notation is used. In this case the relevant constitutive relations are

s

23

¼ 2C

44

ε

23

− e

15

E

2

, s

13

¼ 2C

44

ε

13

− e

15

E

1

,

D

1

¼ 2e

15

ε

23

þ κ

11

E

1

, D

2

¼ 2e

15

ε

13

þ κ

11

E

2

: ð27Þ The displacement M ≡

13

M and potential N ≡

13

N, which appear in (24), are the unique solution of the above mentioned local problem

₁₃

L. In this case Eq. (16) yields

Δ M

^ðγÞ

¼ 0, Δ N

^ðγÞ

¼ 0 in Y

_γ

ð28Þ where Δ is the two-dimensional Laplacian and the contact condi- tions (17) on Γ are written in the form

T

^ð1Þ_s

þT

^ð2Þ_s

¼ 0 on Γ , ‖N‖ ¼ 0, ‖ ðe

₁₅

M

_,δ

−κ

11

N

_,δ

Þn

_δ

‖ ¼ −‖e

15

‖n

1

on Γ , ðC

^ðγÞ₄₄

M

^ðγÞ_,δ

þe

^ðγÞ₁₅

N

^ðγÞ_,δ

Þn

_δ

þC

^ðγÞ₄₄

n

1

¼ ð −1Þ

^γþ1

K

s

C

^ð1Þ₅₅

‖M‖R

⁻¹

on Γ:

ð29Þ Eq. (24) are transformed to area integrals applying Green ´ s theorem. The doubly periodic boundary conditions on Y and the continuity of displacement and potential on Γ leads to

Cⁿ₅₅−iCⁿ45¼〈C55〉þð−1Þ^γC^ðγÞ₅₅ V

Z

ΓM^ðγÞdy₂þiM^ðγÞdy₁þð−1Þ^γe^ðγÞ₁₅ V

Z

ΓN^ðγÞdy₂þiN^ðγÞdy₁,

e

ⁿ₁₅

−ie

ⁿ14

¼ 〈e

15

〉 þð −1Þ

^γ

e

^ðγÞ₁₅

V Z

Γ

M

^ðγÞ

dy

₂

þ iM

^ðγÞ

dy

₁

þð −1Þ

^γ

κ

^ðγÞ11

V Z

Γ

N

^ðγÞ

dy

₂

þiN

^ðγÞ

dy

₁

ð30Þ where summation convention is understood for γ , which run from 1 to 2.

Table 1

Effective properties related to the local problems.

11L 22L 33L 23L 13L 12L 1I 2I 3I

Cⁿ₁₁₁₁ Cⁿ₁₁₂₂ Cⁿ₁₁₃₃

0 0

Cⁿ₁₁₁₂

0 0

eⁿ₃₁₁ Cⁿ₂₂₁₁ Cⁿ₂₂₂₂ Cⁿ₂₂₃₃

0 0

Cⁿ₂₂₁₂

0 0

eⁿ₃₂₂ Cⁿ₃₃₁₁ Cⁿ₃₃₂₂ Cⁿ₃₃₃₃

0 0

Cⁿ₃₃₁₂

0 0

eⁿ₃₃₃

0 0 0

Cⁿ₂₃₂₃ Cⁿ₂₃₁₃

0

eⁿ₁₃₂ eⁿ₂₃₂

0

0 0 0

Cⁿ1323 Cⁿ1313

0

eⁿ₁₃₁ eⁿ₂₃₁

0

Cⁿ₁₂₁₁ Cⁿ₁₂₂₂ Cⁿ₁₂₃₃

0 0

Cⁿ₁₂₁₂

0 0

eⁿ₃₂₁

0 0 0

eⁿ₁₂₃ eⁿ₁₁₃

0

κⁿ11 κⁿ12

0

0 0 0

eⁿ₂₂₃ eⁿ₂₁₃

0

κⁿ₁₂ κⁿ₂₂

0

eⁿ₃₁₁ eⁿ₃₂₂ eⁿ₃₃₃

0 0

eⁿ₃₁₂

0 0

κⁿ₃₃

Methods of potential theory are used to solve (29). Doubly periodic harmonic functions are to be found in terms of the following Laurent and Taylor expansions of harmonic functions:

M

^ð1Þ

ðzÞ ¼ Re z R a

0

þ ∑

^∞

p¼1 0

R

z

p

a

p

þ ∑

^∞

k¼1 0

∑

^∞

p¼1 0

z

R

p

η

kp

a

k

9 =

; , 8 <

:

N

^ð1Þ

ðzÞ ¼ Re z R b

0

þ ∑

^∞

p¼1 0

R

z

p

b

p

þ ∑

^∞

k¼1 0

∑

^∞

p¼1 0

z

R

p

η

kp

b

_k

9 =

; , in Y

1

8 <

:

M

^ð2Þ

ðzÞ ¼ Re ∑

^∞

p¼1 0

c

_p

z

R

p

9 =

; , N

^ð2Þ

ðzÞ ¼ Re ∑

^∞

p¼1 0

d

_p

z

R

p

9 =

; , in Y

2

8 <

: 8 <

:

ð31Þ where

η

kl

¼ − ðkþl−1Þ ! ðk−1Þ !l! R

^kþl

∑

^∞

m¼−∞

∑

^∞

n¼−∞

1

ðmw

₁

þnw

₂

Þ

^kþl

, m

²

þn

²

≠0, kþl4 2 and a

n

,b

n

,c

n

,d

n

are real undetermined coef fi cients; w

1

, w

2

, are the periods of the parallelogram array, respectively (see Fig. 2). The superscript “ o ” next to the summation symbol means that “ p ” runs only over odd integers so that each term in (31) has the same anti- symmetry property as M

^ðγÞ

and N

^ðγÞ

, namely, M

^ðγÞ

ð − zÞ ¼ − M

^ðγÞ

ðzÞ, N

^ðγÞ

ð − zÞ ¼ − N

^ðγÞ

ðzÞ (see more details in the works Bravo-Castillero et al. [10] and Sabina et al. [11]).

The line integrals in (30) and the assumed expansions (31) produce a very simple result as a consequence of the orthogonality of the trigonometric functions, namely

Z

Γ

M

^ð1Þ

dx

2

þiM

^ð1Þ

dx

1

¼ π R a

1

þa

0

þ ∑

^∞

k¼1 o

η

k1

a

k

1 C A , 0

B @

Z

Γ

N

^ð1Þ

dx

2

þiN

^ð1Þ

dx

1

¼ π R b

1

þb

0

þ ∑

^∞

k¼1 o

η

k1

b

k

1 C A , 0

B @

Z

Γ

M

^ð2Þ

dx

2

þiM

^ð2Þ

dx

1

¼ π Rc

1

, Z

Γ

N

^ð2Þ

dx

2

þiN

^ð2Þ

dx

1

¼ π Rd

1

: ð32Þ Replacing (32) into Eq. (30) and taking into consideration the imperfect contact condition (29) we obtain the fi nal expression of the effective coef fi cients,

C

ⁿ₅₅

− iC

ⁿ₄₅

¼ C

^ð1Þ₅₅

ð1 − 2V

2

Π

¹¹

Þ, e

ⁿ₁₅

− ie

ⁿ₁₄

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C

^ð1Þ₅₅

κ

^ð1Þ11

q

ðE

^ð1Þ

− 2V

2

Π

²¹

Þ, ð33Þ where

Π

¹¹

¼ a

1

þEb

1

, Π

²¹

¼ Ea

1

− b

1

, E

^ðγÞ

¼ e

^ð₁₅^γÞ

= ffiffiffiffiffiffiffiffiffiffiffiffiffiffi C

^ð₅₅^γÞ

κ

^ð11^γÞ

q ,

the overbar denotes complex conjugate numbers, the fi ber volume fraction is V

2

¼ π R

²

= V, V ¼ jw

1

jjw

2

jsin θ denotes the area of peri- odic cell. The unknown constants a

1

, b

1

are solutions of the in fi nite systems related to the local problems

₁₃

L, in which only the residue of M

^(γ)

and N

^(γ)

contributes towards C

ⁿ₅₅

, C

ⁿ₄₅

and e

ⁿ₁₅

, e

ⁿ₁₄

. Thus, expressions for a

1

, b

1

are now sought from the system of in fi nite equations

M D ¼ U , ð34Þ

where the vector D

^T

¼ ðx

1

,x

2

,x

3

,x

4

Þ contains the real and imaginary parts of the unknowns a

1

¼ x

1

þix

2

, b

1

¼ x

3

þix

4

and the vector U is given by U

^T

¼ Rð β

21

,0, β

41

,0Þ. The super index T denotes trans- pose and the 4 4-order matrix mðm

nk

Þ is de fi ned by the following matrix form,

M ¼ K þR

²

g −n

1

P

⁻¹

n

2

, ð35Þ K ¼

β

11

0 α

11

0 0 β

11

0 α

11

β

31

0 α

31

0 0 β

31

0 α

31

0 B B B B

@

1 C C C C A ,

J ¼ β

21

h

11

þ h

12

h

21

−h

22

− ðh

21

þ h

22

Þ h

11

−h

12

! α

21

h

11

þ h

12

h

21

−h

22

-ðh

21

þh

22

Þ h

11

−h

12

!

β

41

h

11

þ h

12

h

21

− h

22

− ðh

₂₁

þ h

22

Þ h

11

−h

12

! α

41

h

11

þ h

12

h

21

− h

22

− ðh

₂₁

þ h

22

Þ h

11

−h

12

! 0

B B B B B @

1 C C C C C A

with

h

11

¼ ℜ e δ

1

w

2

−δ

2

w

1

w

1

w

2

− w

2

w

1

, h

12

¼ ℜ e δ

1

w

2

−δ

2

w

1

w

1

w

2

− w

2

w

1

, h

21

¼ ℑ m δ

1

w

2

−δ

2

w

1

w

1

w

2

− w

2

w

1

, h

22

¼ ℑ m δ

1

w

2

−δ

2

w

1

w

1

w

2

− w

2

w

1

, δ

γ

¼ 2 ζ ðw

_γ

= 2Þ, ζ ðzÞ

is the Zeta quasi-periodic Weierstrass function de fi ned as ζ ðzÞ ¼ 1

z þ ∑

^∞

m,n

1 z − T

mn

þ 1

T

mn

þ z T

²_mn

!

, T

mn

¼ mw

1

þnw

2

and the prime over the summation symbol means that the pair (m, n)¼(0, 0) is excluded. The Legendre ’ s relationship links δ

1

, δ

2

and the periods w

1

, w

2

: δ

1

w

2

−δ

2

w

1

¼ π i : The Laurent series expansion of ζ is ζ ðzÞ ¼ ð1 = zÞ −∑

^∞k¼2

c

k

ðz

^2k−1

= 2k − 1Þ, where c

1

¼ 0, c

2

¼ 3 S

4

, c

3

¼ 5S

6

and c

k

¼ ð3 = ð2k þ1Þðk − 3ÞÞ ∑

^km⁻¼²2

c

m

c

_k−m

, k ≥ 4 : The lat- tice S

k

is de fi ned by S

k

¼ ∑

m,n

ðmw

1

þnw

2

Þ

^−k

, m

²

þn

²

≠ 0, k 4 2, S

2

¼ 0 : In particular S

4

and S

6

used in the numerical implementa- tion are reported in Table 1 of Chih-Bing [24] for parallelogram and rhombic cells respectively.

The matrices n

1

, P and n

2

are of in fi nite order and for the numerical implementation it is necessary to truncate to certain order n ∈ℕ: The matrix P ¼

^P⋮^{11 …}⋯ ^P⋮¹ⁿ

Pn1 ⋯ Pnn

!

4n4n

is composed of sub- matrices ð P

^ts

Þ

44

, de fi ned by P

^ts

¼ δ

^ts

K þz

ts

,

K ¼

β

1 2tþ1

0 α

1 2tþ1

0

0 β

1 2tþ1

0 α

1 2tþ1

β

3 2tþ1

0 α

3 2tþ1

0

0 β

3 2tþ1

0 α

3 2tþ1

0 B B B B

@

1 C C C C A ,

zts¼ β2 2tþ1

w1 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w1 2tþ1 2sþ1

! α2 2tþ1

w1 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w1 2tþ1 2sþ1

!

β4 2tþ1

w1 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w1 2tþ1 2sþ1

! α4 2tþ1

w1 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w2 2tþ1 2sþ1 -w1 2tþ1 2sþ1

! 0

BB BB B@

1 CC CC CA ,

w

1kp

¼ ℜ eðw

kp

Þ, w

2kp

¼ ℑ mðw

kp

Þ,

are the real and imaginary parts of the complex number w

kp

¼ ðk þp − 1Þ !

ðk − 1Þ ! ðp − 1Þ ! R

^kþp

ffiffiffiffiffiffi

p kp S

kþp

, k ¼ 2t − 1, p ¼ 2s − 1, t,s ¼ 1,2,3 …:

The matrices n

1

¼ ðn

41

⋯ n

4n

Þ

44n

and n

2

¼

n₁₄

⋮ n_n4

!

4n4

are composed of sub-matrices ðn

4t

Þ

4x4

and ðn

t4

Þ

44

de fi ned by n

_4k

¼ z

2tþ1 1

, n

t4

¼ z

1 2tþ1

respectively. The magnitudes β

1p

, β

2p

, β

3p

, β

4p

, α

1p

, α

2p

, α

3p

, α

4p

are given as follows,

β

1p

¼ 1, β

2p

¼

_1þ^1−χ_χ^{pð1−K−1spÞ}

pð1þK⁻¹_spÞ

, β

3p

¼ 1, β

4p

¼

^Eð1Þ−p

ffiffiffiffiffiffiffi

_χpχtE^ð2Þð1−K⁻¹spÞ E^ð1Þþ^p

ffiffiffiffiffiffiffi

_χpχtE^ð2Þð1þK⁻¹_spÞ

, α

^1p

¼

^{Eð1ÞþχpEð1ÞpK−1s þp}

ffiffiffiffiffiffiffi

_χpχsE^ð2Þ

1þχpð1þK⁻_s¹pÞ

, α

^2p

¼

^{Eð1ÞþχpEð1ÞpK−1s −p}

ffiffiffiffiffiffiffi

_χpχtE^ð2Þ 1þχpð1þK⁻_s¹pÞ

, α

3p

¼

^−1þpp

ffiffiffiffiffiffiffi

_χpχtE^ð1Þ_E^ð2Þ_K−1

s−χt

E^ð1Þþ

ffiffiffiffiffiffiffi

_χ

pχt

p E^ð2Þð1þK⁻¹_s pÞ

, α

4p

¼

^−1þpp

ffiffiffiffiffiffiffi

_χpχtE^ð1Þ_E^ð2Þ_K−1 s þχt

E^ð1Þþ

ffiffiffiffiffiffiffi

_χ

pχt

p E^ð2Þð1þK⁻¹_s pÞ

,

ð36Þ

where

χ

p

¼ C

^ð2Þ₄₄

= C

^ð1Þ₄₄

, χ

t

¼ κ

^ð2Þ11

=κ

^ð1Þ11

:

The limit case of perfect contact condition for piezoelectric antiplane problem is derived as a particular case of (34) – (36) as K

s

-∞ . In this case, the parameters a

1

, b

1

are the same that formula (3.25) page 1475 reported by Bravo-Castillero et al. [10].

The in fi nite system (34) – (36) is used such that it is truncated for obtaining an n n order system. It is interesting to note that the effective properties are monotonic function of order n of the solution of the system. The numerical results converge well to the exact solutions when an adequate order in the solution of the system is chosen as n increase. The truncation order for solving the system increases as the parameters K, χ

ⁿ

and the fi ber volume fraction are high. In the numerical examples the solutions are given for n¼10, because this order of n achieves the require accuracy for the parameters used.

The remaining antiplane problems

₂₃

L,

_α

I ( α ¼1,2) can be solved in analogous form to the aforementioned problem. As a summary, all the effective coef fi cients derived from the antiplane set of local problems can be listed as follows:

C

ⁿ₅₅

− iC

ⁿ₄₅

¼ C

^ð1Þ₅₅

ð1 − 2V

2

H

11

Þ, C

ⁿ₄₅

− iC

ⁿ₄₄

¼ − C

^ð1Þ₅₅

ði þ2V

2

H

12

Þ, e

ⁿ₁₅

− ie

ⁿ₁₄

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C

^ð1Þ₅₅

κ

^ð1Þ11

q ðE − 2V

2

H

21

Þ,

e

ⁿ₁₄

− ie

ⁿ₂₄

¼ − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C

^ð1Þ₅₅

κ

^ð1Þ11

q ðiE þ2V

2

H

22

Þ,

κ

ⁿ11

− i κ

ⁿ12

¼ κ

^ð1Þ11

ð1þ2V

2

H

31

Þ,

κ

ⁿ12

− i κ

ⁿ22

¼ −κ

^ð1Þ11

ði − 2V

2

H

31

Þ ð37Þ where

H

1α

¼ a

1ðα3Þ

þEb

1ðα3Þ

, H

2α

¼ Ea

1ðα3Þ

− b

1ðα3Þ

, H

_3α

¼ Ea

1ðαÞ

− b

1ðαÞ

, E ¼ e

^ð1Þ₁₅

= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C

^ð1Þ₅₅

κ

^ð1Þ11

q ,

the over bar denotes complex conjugate numbers and a

1ðα3Þ

, b

1ðα3Þ

, a

1ðαÞ

and b

1ðαÞ

are solution of the in fi nite systems related to the local problems

13

L,

23

L,

1

I and

2

I

5. Solution of plane local problems

Now, the problem

_ββ

L is considered. We can obtain from (18) the constitutive equations for the plane piezoelectric problem

s

¹¹

¼ C

11 ββ

M

_1,1

þC

12ββ

M

_2,2

, s

²²

¼ C

12 ββ

M

_1,1

þC

11 ββ

M

_2,2

, s

12

¼ C

66

ð

_ββ

M

_1,2

þ

_ββ

M

_2,1

Þ, D

3

¼ e

15 ββ

M

1,1

þe

24 ββ

M

2,2

: ð38Þ

The system of equations related to this plane problem is decoupled into two pure elastic equations s

11,1

þ s

12,2

¼ 0, s

12,1

þ s

22,2

¼ 0, and one electric equation D

3,1

þD

3,2

¼0 with the same unknown functions

_ββ

M

1

,

_ββ

M

2

. The ββ pre-subindices are dropped from all relevant quantities. Therefore, we only need to fi nd the solution of the same problem derived from plane elasticity equations with imperfect contact condition

s

^ðαδ,δ^γÞ

¼ 0 in Y

_γ

, T

^ð1Þ_n

þT

^ð2Þ_n

¼ 0, T

^ð1Þ_t

þT

^ð2Þ_t

¼ 0 on Γ ,

T

^ð_n^γÞ

¼ ð − 1Þ

^γþ1

C

^ð1Þ₆₆

K

n

‖ M

n

‖ R

⁻¹

, T

^ð_t^γÞ

¼ ð − 1Þ

^γþ1

C

^ð1Þ₆₆

K

t

‖ M

t

‖ R

⁻¹

on Γ:

ð39Þ Now, the idea consists to rewrite the mathematical formulation of imperfect contact given by (39) in terms of the potential functions φ

γ

, ψ

γ

.

The methods of a complex variable z in terms of two harmonic functions and the Kolosov – Muskhelishvili complex potentials are applicable. The potentials are related to the displacement and

stress components by means of the formulae 2C

^ð₆₆^γÞ

ðu

^ð₁^γÞ

þiu

^ð₂^γÞ

Þ ¼ χ

^ðγÞ

φ

γ

ðzÞ − z φ

⁼γ

ðzÞ −ψ

γ

ðzÞ, s

^ð11^γÞ

þ s

^ð22^γÞ

¼ 2½ φ

⁼_γ

ðzÞþ φ

⁼γ

ðzÞ,

s

^ð22^γÞ

−s

^ð11^γÞ

þ2i s

^ð12^γÞ

¼ 2½z φ

⁼⁼γ

ðzÞþ ψ

⁼γ

ðzÞ, ð40Þ and χ

^ðγÞ

¼ 3 − 4 ν

^ðγÞ

, ν

^{ð Þγ}

¼ C

^ð₁₂^γÞ

= ðC

^ð₁₁^γÞ

þC

^ð₁₂^γÞ

Þ is the transverse Poisson ' s ratio. The prime denotes a derivative with respect to z. The representation of the complex potentials φ

γ

, ψ

γ

of periods ω

γ

is given in the form

φ

1

ðzÞ ¼ a

0

R zþ ς ðzÞRa

1

þ ∑

^∞

k¼3 n

R

_k

a

_k

∑

^∞

m,n

ðz −β

mn

Þ

^−k

,

ψ1ðzÞ ¼z

Rb₀þςðzÞRb1þQðzÞRa₁þ ∑^∞

k¼3 n R^kb_k∑^∞

m,nðz−βmnÞ^−kþkR^ka_k∑^∞

m,nβmnðz−βmnÞ^−k−1

" #

φ

2

ðzÞ ¼ ∑

^∞

k¼1 n

z

R

k

c

k

, ψ

2

ðzÞ ¼ ∑

^∞

k¼1 n

z

R

k

d

k

, ð41Þ

where the coef fi cients a

0

, b

0

, a

k

, b

k

c

k

d

k

are complex numbers and undetermined, Q(z) is Natanzon ' s function, β

mn

¼mw

1

þnw

2

, w

1

¼1, w

2

¼Re

^iθ

for m,n ∈Z , the asterisk on the sigma symbol means that the double summation excludes the term m¼n¼0.

The double periodicity and quasi-periodicity of these functions leads to

χ

^ð1Þ

a

0

−a

0

¼ ð −A

1

χ

^ð1Þ

a

1

þA

₁

a

1

þA

₂

b

1

ÞR

²

, b

0

¼ ½A

₂

χ

^ð1Þ

a

1

þA

₃

a

1

−A

1

b

1

R

²

, ð42Þ where

A

1

¼ w

1

δ

2

− w

2

δ

1

w

1

w

2

− w

1

w

2

, A

2

¼ w

1

δ

2

− w

2

δ

1

w

1

w

2

− w

1

w

2

, A

3

¼ w

1

P

2

− w

2

P

1

w

1

w

2

− w

1

w

2

, P

_α

¼ 2Q w

_α

2

− w

_α

℘ w

_α

2

, ℘ ðzÞ ¼ −ζ

⁼

ðzÞ :

Using the simple action – reaction principle given by the second and third equation (39) and after some algebraic manipulations of the formulae Kolosov – Muskhelishvili (40) and the series expan- sion of the potential functions (41) we can obtain the following relations between the unknown constants of the above expan- sions:

b

1

¼ 2C B ℜ e

− R

²

A

1

a

1

þ ∑

^∞

k¼1 o

η

k1

a

k

− P

B R γ

2β

, ð43Þ

b

pþ2

¼ p− D

_p

E

p

K

n

K

t

χ

m

ð κ

1

þ1Þ

a

p

− 1 þ B

_p

E

p

K

n

K

t

χ

m

ð κ

1

þ1Þ

∑

^∞

k¼1

o

η

k pþ2

a

k

, ð44Þ

c

1

¼ 1 2ð κ

2

þ1Þ

− C

₁^þ

A

1

R

²

a

1

− C

⁻₁

A

1

R

²

a

1

þC

₁^þ

∑

^∞

k¼1 o

η

k1

a

1

þC

⁻₁

∑

^∞

k¼1

o

η

k1

a

1

þ χ

^ð2Þ

þ1 − 2 β

0

P B

R γ

2j

1 C C C C C A , 0

B B B B B B

@

ð45Þ

c

pþ2

¼ − K

n

K

t

χ

m

ð κ

1

þ1Þ E

p

ðD

p

a

p

þB

p

∑

^∞

k¼1

o

η

k pþ2

a

k

Þ, ð46Þ

d

p

¼ K

n

K

t

χ

m

ð κ

1

þ1Þ C

p

E

p

a

p

þ A

p

E

p

∑

^∞

k¼1 0

η

kpþ2

a

_k

!

þ C

p

ðK

n

− K

t

Þ γ

3j

− A

p

ðK

n

þK

t

Þ γ

3j

þK

n

K

t

C

p

γ

1j

E

p

χ

m

R δ

1p

, ð47Þ where

B ¼ 1 −χ

m

− 2p χ

m

K

n

ðA

⁰_p

Þ

⁻¹

, C ¼ B 1 −κ

²

þ χ

m

ð κ

¹

− 1Þ 2 α

0

þ 2 χ

m

α

0

K

n

, P ¼ B κ

2

− 1

2 α

0

− 4 χ

m

ð1þ γ

4β

=γ

2β

Þ 2 α

0

K

n

, C

₁⁷

¼ 1 þ κ

2

7 χ

m

ð1 þ κ

1

Þþ 2 β

0

C

B ,