Closed-circuit Boundary Conditions Research Articles

An analytical solution for vibration analysis of sandwich plates reinforced with graphene nanoplatelets

In this study, the free vibration of a composite sandwich plate reinforced with graphene nanoplatelets (GPLs) enclosed by piezoelectric layers is investigated using an analytical solution. In the framework of the first-order shear deformation plate theory, multilayer functionally graded graphene platelets-reinforced composite plate is assumed. Applying modified Halpin–Tsai model and rule of mixtures, the effective Young’s modulus, mass density and Poisson’s ratio of nanocomposites are predicted. In each individual layer, the weight fraction of GPL nanofillers illustrates a layer-wise variation along the thickness direction either uniformly or non-uniformly GPLs dispersed. Based on Maxwell’s equation, the electric potential in a piezoelectric layer is considered for open and closed circuit boundary conditions. Coupled governing equations of motion and boundary conditions are derived by using the Hamilton’s principle. Four auxiliary scalar functions are introduced to decouple the governing equations of motion and boundary conditions, which are solved analytically by employing Levy-type boundary conditions. The effects of GPLs weight fraction, GPLs distribution patterns, number of layers and aspect ratio are examined in detail. The results show that the best way to predict the most effective reinforcement is to distribute more GPLs with a larger surface area near the top and bottom surfaces of the plate. Besides, adding a small amount of GPLs as reinforcing nanofillers can significantly improve the stiffness of the plate.

Vibration attenuation and dynamic control of piezolaminated plates with coupled electromechanical actuation

Vibration attenuation and dynamic control of the piezolaminated plate actuated with coupled electromechanical loading are aimed and achieved in the present work. Efficient finite element methodology with higher-order shear deformation theory is assisted to blend the effects of shear strains in the computational model. An isoperimetric eight-noded rectangular element is implemented through linear electric potential distribution along the thickness to set up the relations. Coupled electromechanical loading is considered in the formulation to consider the piezoelectric sensing and actuation mechanism. Piezolaminated plates with both cross-ply and angle-ply orientation of bonded substrate lamina are considered in the analysis. Vibration responses of multilayered composite plates with surface-mounted piezolaminates are evaluated for open-circuit and closed-circuit boundary conditions of the electric field. Results obtained from the analysis are well revealed for frequencies of various modes of simply supported piezolaminated plates. Dynamic vibrational response and effect of the control gain (Gv) in terms of velocity feedback on the dynamic characteristics of a piezoelectric laminated plate are examined. Another aim of proposing this FE model is to demonstrate the effect of placement of fibers in a different orientation for the PVDF layer to achieve active damping, and attenuation of vibrations of the laminated composite plate is also achieved and demonstrated.

Buckling analysis of orthotropic smart laminated nanoplates based on the nonlocal continuum mechanics using third-order shear and normal deformation theory

In this article, the nonlocal buckling behavior of biaxially loaded graphene sheet with piezoelectric layers based on an orthotropic intelligent laminated nanoplate model is studied. The nonlocal elasticity theory is used in the buckling analysis to show the size scale effects on the critical buckling loads. The electric potential in piezoelectric layers satisfies Maxwell’s equation for either open- or closed-circuit boundary conditions. Based on the third-order shear and normal deformation theory, the nonlinear equilibrium equations are obtained. In order to obtain the linear nonlocal stability equations, the adjacent equilibrium criterion is used. The linear nonlocal governing stability equations are solved analytically, assuming simply supported boundary condition along all edges. To validate the results, the critical buckling loads are compared with those of molecular dynamics simulations. Finally, the effects of different parameters on the critical buckling loads are studied in detail. The results show that by increasing the nonlocal parameter, the critical buckling load decreases. The piezoelectric effect increases the critical buckling load for both open- and closed-circuit boundary conditions. For open-circuit boundary condition, the variation in the critical buckling load is due to the stiffness and piezoelectric effects, but for closed circuit, it is due to the stiffness effect only. Also, the critical buckling load for open circuit is bigger than that of closed one.

Free vibration analysis of micro-magneto-electro-elastic cylindrical sandwich panel considering functionally graded carbon nanotube–reinforced nanocomposite face sheets, various circuit boundary conditions, and temperature-dependent material properties using high-order sandwich panel theory and modified strain gradient theory

In this article, based on high-order sandwich panel theory and modified strain gradient theory, free vibration analysis of a micro-magneto-electro-elastic sandwich panel with a transversely flexible core and functionally graded carbon nanotube–reinforced nanocomposite face sheets is investigated. Also, the influences of temperature-dependent material properties and various circuit boundary conditions such as open and closed are considered in this study. Carbon nanotubes are arranged in longitudinal direction inside polyvinylidene fluoride matrix with various functionally graded (FG) distributions such as uniform, FG-V, FG-A, FG-X, and FG-O in the face sheets. The generalized rule of mixture is employed to predict mechanical, electrical, magnetic, and thermal properties of micro-sandwich composite panel. The classical shell theory and an elasticity high-order theory are used for the face sheets and the core, respectively. Then, the governing equations of motion are derived using Hamilton’s principle. In this article, the influences of the volume fraction, the various distributions of carbon nanotubes, the multi-physical fields, open- and closed-circuit boundary conditions, the material length scale parameters, different face sheet and core thicknesses, and temperature changes on the natural frequency are investigated, and the obtained results show that these influences play an important role in the natural frequencies and can be used in order to prevent the resonance phenomenon and also for manufacturing process design and optimization of micro-magneto-electro-elastic composite sandwich cylindrical panels.

Thermal buckling of FGEM plates integrated with surface-bonded piezoelectric layers and temperature-dependent material properties

ABSTRACTBased on Reissner’s mixed variational theorem (RMVT), a unified formulation of finite layer methods (FLMs) is developed for the quasi three-dimensional (3-D) thermal buckling analysis of simply-supported, sandwich piezoelectric plates embedded with a functionally graded elastic material (FGEM) core, the material properties of which are considered to be thickness- and temperature-dependent. The plate is subjected to a uniform temperature change and with open-/closed-circuit boundary conditions on the lateral surfaces. A 3-D linear buckling theory is used, in which a set of membrane stresses is assumed to exist just before buckling occurs, and these membrane stresses are determined using a set of predefined 3-D deformations for the pre-buckling state. The material properties of the FGEM core are assumed to obey the power-law distributions varying through the thickness coordinate of the core according to the volume fractions of the constituents. The effective material properties are estimated using the rule of mixtures and Mori-Tanaka’s model. The accuracies and convergence rates of the FLMs with various orders, as used for expanding the elastic and electric variables in the thickness direction, are assessed by comparing their solutions with the exact 3D and accurate two-dimensional ones available in the literature.

Topology optimization of piezoelectric nanostructures

We present an extended finite element formulation for piezoelectric nanobeams and nanoplates that is coupled with topology optimization to study the energy harvesting potential of piezoelectric nanostructures. The finite element model for the nanoplates is based on the Kirchoff plate model, with a linear through the thickness distribution of electric potential. Based on the topology optimization, the largest enhancements in energy harvesting are found for closed circuit boundary conditions, though significant gains are also found for open circuit boundary conditions. Most interestingly, our results demonstrate the competition between surface elasticity, which reduces the energy conversion efficiency, and surface piezoelectricity, which enhances the energy conversion efficiency, in governing the energy harvesting potential of piezoelectric nanostructures.

Analytical solution for free vibrations of moderately thick hybrid piezoelectric laminated plates

An analytical solution for free vibrations of a hybrid rectangular plate composed of a transversely isotropic, homogeneous and linear elastic core and face sheets made of a linear piezoelectric material is derived by assuming that the plate deformations are governed by the Mindlin plate theory. The electric potential in a piezoelectric layer satisfies Maxwell's equation and either open circuit or closed circuit boundary conditions on its major surfaces. For the hybrid plate coupled governing equations obtained from the Hamilton principle are decoupled by introducing four auxiliary scalar functions, and the Levy type analytical solution for free vibrations is derived. Plate frequencies as a function of the piezoelectric layer thickness and the plate aspect ratio are presented and discussed. It is found that the electric boundary conditions on major surfaces of the piezoelectric layers and the aspect ratio of the hybrid plate noticeably influence its frequencies. Significant contributions of the work include proposing the four scalar functions to uncouple the governing equations, providing an analytical solution for frequencies of the hybrid plate, and delineating effects of boundary conditions as well as of aspect ratio of the plate.

Boundary conditions in periodic density functional calculations of insulating materials

Ewald's method yields internal electric fields in a polar material consisting of local and macroscopic components. The macroscopic component is a priori undefined. It depends on the boundary conditions imposed at the surface of the macroscopic medium. The boundary conditions ordinarily used in periodic electronic-structure calculations correspond to assuming that the modeled material is embedded in a metal connected to the ground and are equivalent to considering the macroscopic components of the internal electric fields to be null. These boundary conditions are commonly referred to as closed-circuitboundary conditions. In this study, we discuss and use an alternative type of boundary conditions, suitable for describing the most common physical situation of a polar material with the surface exposed to the vacuum, that is, under open-circuit conditions. We implement these boundary conditions in a periodic density functional approach and use the resulting energy scheme to study NaCl, MgO, and BaO. A comparison with the experiments shows that periodic density functional calculations employing the open-circuit boundary conditions give an excellent description of both dynamical and dielectric properties of polar materials.

A Coupled Zig-Zag Third-Order Theory for Piezoelectric Hybrid Cross-Ply Plates

A new zig-zag coupled theory is developed for hybrid cross-ply plates with some piezoelectric layers using third-order zig-zag approximation for the inplane displacements and sublayer wise piecewise linear approximation for the electric potential. The theory considers all electric field components and can model open and closed-circuit boundary conditions. The deflection field accounts for the transverse normal strain due to the piezoelectric d33 coefficient. The displacement field is expressed in terms of five displacement variables (which are the same as in FSDT) and electric potential variables by satisfying exactly the conditions of zero shear stresses at the top and bottom, and their continuity at layer interfaces. The governing equations are derived from the principle of virtual work. Comparison of the Navier solutions for the simply-supported plates with the analytical three-dimensional piezoelasticity solutions establishes that the present efficient zig-zag theory is quite accurate for moderately thick plates.

An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams

An efficient one-dimensional model is developed for the statics of piezoelectric sandwich beams. Third-order zigzag approximation is used for axial displacement, and the potential is approximated as piecewise linear. The displacement field is expressed in terms of three primary displacement variables and the electric potential variables by satisfying the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces. The deflection field accounts for the piezoelectric transverse normal strain. The governing equations are derived using a variational principle. The present results agree very well with the exact solution for thin and thick highly inhomogeneous simply supported hybrid sandwich beams. The developed theory can accurately model open and closed circuit boundary conditions.