Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 281 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. V ARIATIONAL FORMULATION FOR THE THREE - DIMENSIONAL PROBLEM OF LINEAR PIEZOELECTRICITY et  be an open connected domain of 3  with smooth boundary  made of two parts 1 and   2   in the mechanical sense, and of 3   and 4   in the electrical one. These parts of  represent portions of regular surfaces with smooth common boundary, respectively. Moreover,  may be divided into two parts by a smooth surface . S  Figure 1 : Representation of the open  . In the framework of linear piezoelectricity, the elastic and electric effects are coupled by the constitutive equations: k ij ijkl kij k j u a e E x      (1) k i ij j ikl l u D E e x      (2) where { } i u u  is the elastic displacement, { } ij    is the symmetric stress tensor, { } i E E  is the electric field vector, and { } i D D  is the electric displacement vector, with (i, j, k, l) = (1, 2, 3). We now assume that the elastic coefficients at zero elastic field ijkl a , the piezoelectric coefficients kij e and the dielectric constants ij  at vanishing strain satisfy the following symmetry and positivity properties: ; ; ijkl jikl klij kij kji ij ji a a a e e       (3) 0 , , 0 : ij ij ji ijkl kl ij ij ji e e e a e e e e       (4) L