Finite elements for nonlinear free vibrations analysis of smart laminates subjected to in-plane loadings

Abstract

Advanced composites, able to provide multi-functional capabilities besides the traditional structural functions, has been gaining attention in many technological fields. This inherent coupling of different physical fields can be exploited in transducer applications, structural health monitoring, vibration control, energy harvesting and other applications. Magneto-electro-elastic (MEE) composite materials are attracting increasing consideration as they couple mechanical, electrical and magnetic fields and this makes them particularly suitable for smart applications. Generally, single-phase materials exhibit either piezoelectric or piezomagnetic behavior and no direct magneto-electric coupling is observed. However, the full magneto-electro-elastic coupling can be obtained by using composites with both piezoelectric and piezomagnetic phases that provide the magneto-electric effect through the elastic field. These MEE composites are obtained in the form of multi-phase materials, i.e. piezoelectric and piezomagnetic particles and/or fibers, or in the form of laminated structures, with piezoelectric and piezomagnetic layers stacked to achieve the desired coupling effects. Multilayered configurations appear to be more effective than bulk composites. For the analysis and design of MEE structures, reliable and efficient modeling tools are required. Analytical solutions are available for simple configurations and, actually, numerical approaches need to be deployed for complex analyses. Fully-coupled 3D finite element solutions for multilayered plates and shells present very high computational costs; 2D laminated plate theories and the corresponding finite element solutions have been developed with the aim of reducing the analysis effort while preserving a suitable level of accuracy. In the framework of 2D plate theories, finite elements solutions based on equivalent-single-layer or layer-wise modeling have been proposed implementing different order theories. Recently, an equivalent single-layer approach for multilayered MEE plates and its finite element solution have been proposed by the author, who developed an effective purely mechanical plate model as result of the condensation of the electro-magnetic state to the mechanical variables. This model was systematically extended to refined equivalent-single-layer and layer-wise plates theories approaching the problem through a suitable application of the Carrera Unified Formulation (CUF). Finite element solutions for magneto-electro-elastic multilayered plates obtained by theories with different expansion order have been presented. In the present work, a unified framework based on CUF is presented to develop layer-wise and equivalent-single-layer plate models for the nonlinear free vibrations analysis of MEE laminates. Variable kinematics with von Karman strains is assumed and approximated by standard isoparametric finite elements. Under the assumption of quasi-static behavior of the electromagnetic fields, the electromagnetic state of each single layer is preliminary determined by solving the corresponding governing equations coupled with the proper interface continuity and external boundary conditions. This allow condensing the electromagnetic state into the plate kinematics and the layer governing equations are inferred by the principle of virtual displacements. This approach identifies effective mechanical layers, which are kinematically equivalent to the original smart layers. These effective layers are characterized by stiffness and inertia load properties, which consider the multifield coupling effects as their definitions involve the electromagnetic coupling material properties. The layers equations are finally assembled enforcing the mechanical interface conditions. This allows obtaining the smart plate FE resolving system, which involves mechanical nodal variables only. Numerical results are presented.