Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 280 Theoretical model of homogenized piezoelectric materials with small non-collinear periodic cracks Jean-Marie Nianga, Driss Marhabi Pôle de Recherche « Structures & Matériaux », Hautes Etudes d’Ingénieur – (HEI), 13 rue de Toul, 59046 Lille Cedex, France jean-marie.nianga@yncrea.fr A BSTRACT . 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. K EYWORDS . Piezoelectric material; Asymptotic expansions; Homogenization; Variational formulation; Periodic cracks . Citation: Nianga, J.-M., Marhabi, D., Theoretical model of homogenized piezoelectric materials with small non- collinear periodic cracks, Frattura ed Integrità Strutturale, 42 (2017) 280-292. Received: 01.02.2017 Accepted: 04.07.2017 Published: 01.10.2017 Copyright: © 2017 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. I NTRODUCTION he piezoelectric materials are used in an increasing way in technological applications [1-3]. 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. T