Hierarchic Plate and Shell Theories with Direct Evaluation of Transverse Electric Displacement

A mixed variational statement for the analysis of layered structures under the effect of mechanical and electrical fields is proposed in this paper to develop finite plate elements what permits direct evaluation (that is “a priori”) of transverse electrical displacement Dz. The original Reissner Mixed Variational Theorem RMVT is modified to account “only” for interlaminar continuous Dz. Continuity of mechanical variables, such as transverse shear and normal stress components, is discarded to provide a simple “Electrical” modified RMVT, here denoted RMVT-Dz. Implementation are made via Carrera Unified Formulation. The applications of the proposed approach is demonstrated by comparison with classical formulations based on the Principle of Virtual Displacements as well as to available 3D and analytical solutions. Smart systems can be considered the candidates for next generation structures in aerospace vehicles as well as for some advanced products in the automotive and ship industries. Piezoelectric materials are extensively used in this framework. These materials are characterized by the so-called “direct” and “inverse effect”: an applied electrical potential induces mechanical stresses and vice-versa. Such an electro-mechanical coupling permits one to build closed-loop control systems in which piezo-materials play the role of both actuators and sensors. An intelligent structure can therefore be built in which, for instance, deformations or vibrations are reduced by appropriate control laws. An appropriate use of piezoelectric materials, however requires an accurate description of the electrical and mechanical fields in the constitutive layers. The present paper focuses on the computational, finite element, electro-mechanical two-dimensional modelings of smart structures embedding piezo layers. Piezoelectric plates appear as multilayered structures. Very often, piezoelectric layers are embedded in laminated structures made of anisotropic composite materials. The importance of appropriate modelings of piezoelectric plates is clearly displayed by the large number of papers that have been published over the last two decades. Among the available review papers, those by Saravanos and Heyliger, 1 and Wang and Yan 2 are herein mentioned. A short review of some of the latest contributions to the FE analysis of piezoelectric plates follows. A finite element that includes a FSDT description of displacements and layer-wise form of the electric potential was developed by Sheik et al. 3 The numerical, membrane and bending behavior of FEs which are based on FSDT was analyzed by Auricchio et al. 4 in the framework of a suitable variational formulation. The third order theory of HOT type was applied by Thornburg and Chattopadhyay 5 to derive finite elements that take into consideration the electro-mechanical coupling. Similar elements have more recently been considered by Shu. 6 The extension of the third order Zig-Zag Ambartsumian multilayered theory to finite analysis of electromechanical problems has been proposed by Oh and Cho. 7 An extension to piezoelectricity of numerically efficient plate/shell elements based on the Mixed Interpolation of Tensorial �