A Generalised Modelling of Piezoelectric Composite Shells and Plates using Variational Asymptotic Method

Abstract

In several fields of engineering, researchers are actively pursuing the possibility to use smart materials such as piezoelectric composites in order to make the structure smart. Piezoelectric materials have the capability to sense and react to external stimuli and thus provide engineers with a vast array of its application and build structures with it for better performances. In order to fully exploit the potential of the smart materials, efficient yet accurate models are indispensable for design and analysis of these structures. This present work proposes a generalised formulation for the electro-mechanical analysis of multilayered plates/shells embedding piezoelectric composite layers. Many engineering components are at/curved panels which can be analysed using the two-dimensional (2D) plate or shell models. Hence, it is essential to have computationally efficient yet accurate models are necessary for design and analysis of these structures. A strain energy function is obtained for a fully coupled Reissner-Mindlin model for piezoelectric composite plates and shells with some surfaces parallel to the reference surface coated with electrodes. The three-dimensional strain energy is based on geometrically nonlinear elasticity theory. We have used variational asymptotic method as the tool to divide the actual three dimensional electromechanical problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate and shell analysis, where thickness and strain serve as the small parameters. The through-the-thickness analysis serves by providing a constitutive model for the plate/shell analysis. Moreover, the through-the-thickness analysis derives relationships between the three-dimensional field variables in terms of global responses derived from the plate/shell analysis. In spite of the simple form for the plate/shell strain energy, it caters the effect of large deection of the system by applying no constraints on the magnitudes of displacement and rotation measures. Having this generality in the kinematics, no more variables are involved than in Reissner-Mindlin plate theory. However, the asymptotically correct analytical expressions for the constitutive model of plate/shell and the three-dimensional field variables provide valuable insight to the nonlinear sensitivity of these variables to the stiffness parameters and global responses. The present two-dimensional plate/shell theory can be implemented into the FEM solver (we have used ABAQUSr) to derive global displacements. We use the displacement measures, derived using FEM, in the calculation of geometrically exact two-dimensional strain measures. After solving the plate/shell problem we substitute these results back into the relations for calculation of three-dimensional displacements and strains throughout the thickness of the plate/shell. The proposed model is as simple as an equivalent single layer, first-order shear deformation theory with accuracy comparable to higher-order layer-wise theories. We have used numerical examples to validate the present asymptotically correct analytical model with Classical Lamination Theory (CLT), First-Order Shear Deformation Theory, equivalent single layer theories and layerwise theories. Moreover, we establish its advantage over the prohibitive computational cost of 3D FEM