Issue 29

C. Maruccio et alii, Frattura ed Integrità Strutturale, 29 (2014) 49-60; DOI: 10.3221/IGF-ESIS.29.06 50 materials. From a commercial perspective the term "nano" describes a diameter of the fiber below one micron. However, the most important properties of these materials tend to manifest at a scale below 500 nanometers. Materials characterized by nanofibers at the microlevel show high surface area and superior mechanical, electric, magnetic properties [4]. In this framework, polyvinylidene fluoride (PVDF) is a fluoropolymer known for its strength, chemical resistance, thermal resistance and piezoelectric properties. In particular PVDF nanofibers are suitable for numerous applications such as filtration, coating, sensors, and energy generators. From the chemical point of view, the PVDF material is built joining chains of CH2CF2, where C indicates the carbon, H the hydrogen and F the fluorine atoms. It is produced in large thin clear sheets and through a stretching and poling process it is possible to give piezoelectric properties to the resulting thin layer. The stretch direction is the direction along the sheet in which most of the carbon chains run. The hydrogen atoms, which have a net positive charge, and the fluorine atoms, which have a net negative charge, end up on opposite sides of the sheet. This creates a pole direction that is either oriented to the top or bottom of the sheet. When an electric field E is applied across the sheets, they either contract in thickness and expand along the stretch direction or expand in thickness and contract along the stretch direction depending on which way the field is applied. This is due to the physical nature of the positive hydrogen atoms attracted by the negative side of the electric field and repelled by the positive side of the electric field. Scanning electron micrographs of PVDF layer architectures at the microscale show that, depending on the process parameters, the resulting microstructure can range from a fully random geometry to excellent mutual alignment of fibers [5]. Several analytical and computational multiscale approaches have been developed in order to predict the macroscale properties of heterogeneous materials at the lower scale(s), [6-11]. Although most of these efforts have been devoted to continuum mechanics [12, 13], some applications to multiphysics problems are also available [14, 15]. A few of these concern electromechanically coupled problems such as in the case of piezoelectricity [16]. Moreover, although several macroscale formulations were developed for piezoelectric shell elements [17-19], computational homogenization of shells is only recently receiving major attention [20, 21]. To the best of our knowledge, such approaches have not yet been proposed for piezoelectric shells. In this framework, the objective of this paper is to model the behaviour of the aforementioned PVDF sheet by a multiscale and multiphysics approach. This requires the definition of a representative volume element (RVE) at the microscale, the formulation and solution of a microscale boundary value problem (BVP), and the development of a suitable micro-macro scale transition. In the first part of the paper a microscale RVE element is defined and some details about the formulation of suitable electromechanical contact laws to describe the interaction among the fibers are provided. The second part describes the kinematic behaviour of a piezoelectric shell and the multiscale approach. Finally, based on the presented framework, several RVE geometries are analysed and the homogenized coefficients determined. M ICROSCALE FORMULATION AND HOMOGENIZATION PROCEDURE he microscopic length scale of the RVE (micrometers) is several orders of magnitude smaller than the macroscopic dimensions (centimeters). Hence, two models are developed: one at the micro-level and one at the macro-level. The problem is how to couple these models and which boundary conditions to apply to the micro- model. At the microscale, the RVE consists in piezoelectric polymer fibers that feature a linear piezoelastic constitutive behavior and are subjected to electromechanical contact constraints. The governing equations are the Navier equations and the strain-displacement relations for the mechanical field and Gauss and Faraday laws for the electrostatic field. Moreover, the constitutive equations read: ) )      ij ijkl kl kij k i ikl kl ik k a T C S e E b D e S E  (1) where ijkl C , ikl e , and  ik  are respectively the elastic, piezoelectric, and permittivity constants, whereas , ij ij S T are the strain and stress components and , i i D E are the electric displacement and the electric field components, respectively. The interaction among the fibers is described defining a 3D electromechanical frictionless contact law and implementing an element with the following main characteristics:  the contact formulation is based on the master-slave concept;  Bézier patches are used for smoothing of the master surface;  the impenetrability condition is extended to the electromechanical setting by imposing equality of the electric T