Abstract
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.
Highlights
The piezoelectric materials are used in an increasing way in technological applications [1,2,3]
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]
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]
Summary
The piezoelectric materials are used in an increasing way in technological applications [1,2,3]. [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 They analyzed the interactions of a row of cracks periodically located in a piezoelectric material layer. 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. For the existence and uniqueness of the solution of problem (VP), see [9]
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