Theoretical model of homogenized piezoelectric materials with small non-collinear periodic cracks

An analytical model for the homogenization of a piezoelectric material with small periodic fissures is proposed on the basis of the method of asymptotic expansions for the elastic displacement, the electric scalar potential and the test functions. Starting from the variational formulation of the three-dimensional problem of linear piezoelectricity, we have at first obtained that concerning a cracked piezoelectric structure, before the implementation of homogenized equations for a piezoelectric structure with a periodic distribution of cracks. It then follows, the characterization of the homogenized law between the mechanical strain and the electric potential, on one hand, and the mechanical stress and the electric displacement, on the other hand. Contrary to the previous investigations, the focus of this paper is the development of a mathematical model taking the non-parallelism of cracks into account.


INTRODUCTION
he piezoelectric materials are used in an increasing way in technological applications [1][2][3]. Among the numerous problems which can arise, there is that concerning the global estimation of the homogenized characteristics of non-homogeneous materials, in particular those presenting a periodic distribution of singularities. Significant efforts had been made to the study of periodical cracks in linear piezoelectricity, through an extension to piezoelectric materials of the modeling of periodic cracks in elastic materials [4,5]. Gao and al. [6] studied, in terms of the Parton assumption and Stroh formalism, the problem of a half-infinite crack in piezoelectric media with periodic crack; reducing it to Hilbert one and getting therefore the closed-form solutions in the media and inside the cracks. Wang and al. [7] provided a theoretical treatment of the dynamic interaction between cracks in a piezoelectric medium under anti-plane mechanical and in-plane electrical incident wave. They used Fourier transform to study the dynamic electromechanical behavior of a single crack, and solved the obtained singular integral equations by Chebyshev polynomials. The single crack solution was then implemented into a pseudo-incident wave method to account for the interaction between cracks.
Somme years more-late, Han and al. [8] obtained the development of a mathematical model to predict the length scale for the spacing of transverse cracks forming in a piezoelectric material subjected to a coupled electro-mechanical external loading condition. In particular, they analyzed the interactions of a row of cracks periodically located in a piezoelectric material layer. Although, one of the remaining problems that need to be treated is that of a periodic array of non-collinear cracks. So, the present paper provides a theoretical model of homogenized piezoelectric materials with small non-collinear periodic cracks through an extension of previous works [9] and [10]. It is organized as follows: Section 2 describes the variational formulation for the three-dimensional problem of linear piezoelectricity. Section 3 develops a variational formulation for the problem of a fissured piezoelectric structure. In Section 4, are presented the homogenized problem of a piezoelectric material with small periodic cracks. Section 5 is then devoted to the formulation of the homogenized local problem in the homogenization period. The analysis of the relationship between the strain and the electric potential on one hand, and the stress and the electric field secondly, is presented in Section 6, just above the conclusion. In the framework of linear piezoelectricity, the elastic and electric effects are coupled by the constitutive equations: Proof. (19) 1 is obtained by multiplying (11) par a test function v i and by integrating by parts; taking into account the boundary conditions (7) and (8). By analogy, we obtain (19) 2 , by multiplying (12) par a test function  and by integrating by parts; taking into account the boundary conditions (9) and (10). The coefficients ijkl a are assumed to be continuous on S  . For the existence and uniqueness of the solution of problem (VP), see [9].

VARIATIONAL FORMULATION FOR THE PROBLEM OF A FISSURED PIEZOELECTRIC STRUCTURE
e now consider a piezoelectric structure containing a closed crack C, i.e.
where C is the closure of C, and where C  is assumed to be smooth. Let us introduce the open subset , C  verifying: The local equations of linear piezoelectricity for a fissured piezoelectric structure can then be written as follows [10]: where N and n represent the unit normal to C, outer to its side 1, see (  These relations express a compression on C according to (33), as well as the normality of the force which acts; with an opposition between the action and the reaction, according to (32). Consequently, the variational formulation (FVP) for the problem of such a fissured piezoelectric structure can be stated as follows: And where a, b, c, and d are bilinear forms on Proposition2. Problem (FVP) is equivalent to Eqs. (26) to (34).
The proof is analogous to that of proposition 1, by taking into account Eqs. (29) and (31).

HOMOGENIZED EQUATIONS-FORMAL EXPANSION
e now consider a linear piezoelectric plate with a   periodic distribution of fissures, so that, the period Y of 3 , R admits a smooth fissure C verifying: The fissured material denoted by C  is then defined as follows: And we assume that, there is no fissure intersecting the boundary  of the open .  Introducing the following spaces: The corresponding variational formulation ( FVP ) of such a piezoelectric problem in , C  is then defined as follows: In order to study the asymptotic behavior of the solution when  tends to zero, we use the classical following expansions, both for the unknown and the test functions: Introducing the following spaces: Comparing (47)-(50) with (46), we get the following relations: with  represents the operator average defined on any Y-periodic function f(y) by: 1 In the particular case where 1 1 1

,
v u and     we get: Consequently, the corresponding homogenized equations in which there are no more fissures then follow: et us choose the fields 1 1 v and  as:

HOMOGENIZED LOCAL PROBLEM IN THE PERIOD Y
where ( ) D  represents the set of the infinitely derivable functions with compact support in  .