Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers

Nonlinear vibration analysis of thin circular pre-stressed functionally graded (FG) plate integrated with two uniformly distributed piezoelectric actuator layers with an initial nonlinear large deformation are presented in this paper. Nonlinear governing equations of motion are derived based on classical plate theory (CPT) with von-Karman type geometrical large nonlinear deformations. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectrically actuated circular FG plate are derived. An exact series expansion method is used to model the nonlinear electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages are studied and their nonlinear effects are evaluated. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FG plate on the dynamic characteristics of the structure.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

Bending analysis of functionally graded annular sector plates by extended Kantorovich method

The small deflection equation for isotropic and non-homogenous thin annular sector subject to transverse loading is derived in polar coordinate. The Poisson’s ratio of the plate is assumed to be constant, but the Young’s modulus varies continuously through the radial direction according to the power-law function. A closed-form solution with fast convergence rate is obtained using the extended Kantorovich method and the classical theory of thin plates (Kirchhoff theory). The validity and accuracy of results are examined by resolving the problem using commercial finite element code ABAQUS and a good agreement between them is obtained for either deflection or stress resultants.     

Hybrid damping of smart, functionally graded plates using piezoelectric, fiber-reinforced composites

This paper deals with the investigation of active, constrained layer damping (ACLD) of smart, functionally graded (FG) plates. The constraining layer of the ACLD treatment is considered to be made of a piezoelectric, fiber-reinforced composite (PFRC) material with enhanced effective piezoelectric coefficient that quantifies the in-plane actuating force due to the electric field applied across the thickness of the layer. The Young's modulus and the mass density of the FG plates are assumed to vary exponentially along the thickness of the plate, and the Poisson's ratio is assumed to be constant over the domain of the plate. A finite-element model has been developed to model the open-loop and closed-loop dynamics of the FG plates integrated with two patches of ACLD treatment. The frequency response of the plates revealed that the active patches of ACLD treatment significantly improve the damping characteristics of the FG plates over the passive damping. Emphasis has been placed on investigating the effect of variation of piezoelectric fiber angle in the constraining layer of the ACLD treatment on the attenuating capability of the patches. The analysis also revealed that the activated patches of the ACLD treatment are more effective in controlling the vibrations of FG plates when the patches are attached to the surface of the FG plates with minimum stiffness than when they are attached to the surface of the same with maximum stiffness.     

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Love waves in a smart functionally graded piezoelectric composite structure

The propagation of Love waves in a smart functionally graded piezoelectric structure is analyzed by applying elastic wave theory. There is an additional functionally graded layer between the piezoelectric layer and the substrate in this smart structure. When the piezoelectric and dielectric constants vary individually in a functionally graded layer, the asymptotic solutions of Love waves are obtained by applying the WKB method and solving the fourth order differential equation with variable coefficients. The effects of gradient variation on the phase velocity and the coupled electromechanical factor are discussed in detail. The analysis shows that the number of vibration modes is greater than that in the non-graded layer structure, and the coupled electromechanical factor increases with the increase of piezoelectric constant graded variation. Presented results are useful for the improvement of properties of surface acoustic wave (SAW) devices.     

Benchmark solution for free vibration of functionally graded moderately thick annular sector plates

In the present article, an exact analytical solution for free vibration analysis of a moderately thick functionally graded (FG) annular sector plate is presented. Based on the first-order shear deformation plate theory, five coupled partial differential equations of motion are obtained without any simplification. Doing some mathematical manipulations, these highly coupled equations are converted into a sixth-order and a fourth-order decoupled partial differential equation. The decoupled equation are solved analytically for an FG annular sector plate with simply supported radial edges. The accurate natural frequencies of the FG annular sector plates with nine different boundary conditions are presented for several aspect ratios, some thickness/length ratios, different sector angles, and various power law indices. The results show that variations of the thickness, aspect ratio, sector angle, and boundary condition of the FG annular sector plates can change the vibration wave number. Also for an FG annular sector plate with one free edge, in opposite to the other boundary conditions, the natural frequency decreases with increasing the aspect ratio for small aspect ratios. Moreover, the mode shape contour plots are depicted for an FG annular sector plate with various boundary conditions. The accurate natural frequencies of FG annular sector plates are presented for the first time and can serve as a benchmark solution.     

Nonlinear thermoelastic analysis of thick-walled functionally graded piezoelectric cylinder

This research addresses the nonlinear behavior of a functionally graded piezoelectric (FGP) cylinder under thermal, mechanical and electrical loads. Except the Poisson ratio, the other thermal, mechanical and electrical properties vary continuously along the thickness direction. The geometric nonlinearity was considered in strain–displacement relation. The present paper proposes a novel analytical method for estimating the response of the systems of nonlinear differential equations. The proposed method is modification and combination of Adomian’s decomposition and successive approximation methods. The governing nonlinear differential equations were solved using the proposed analytical method. The nonlinear solutions were presented for different values of nonhomogenous index. For validation of the results and justifying the necessity of the present study, the present results were compared with results obtained using the linear analysis. Due to using the functionally graded cylinder in serious environments such as chemical and weapon equipments or adjustable and precise condition, a nonlinear analysis for these materials is necessary. A nonlinear analysis improves the results significantly. A maximum of 5% improvement for electric potential and 9% for radial displacement were attained employing this nonlinear analysis rather than a linear analysis. These considerable improvements can present a confidence and adjustable tool for manufacturing and design of sensors and actuators with better accuracy.     

Optimal shape control of functionally graded smart plates using genetic algorithms

This paper deals with optimal shape control of functionally graded smart plate containing patches of piezoelectric sensors and actuators. The genetic algorithm (GA) is designed to search for optimal actuator voltage and displacement control gains for the shape control of the functionally graded material (FGM) plates. The work extends the earlier finite element formulations of the two leading authors, so that it can be readily treated using genetic algorithms. Numerical results have been obtained to study the effect of the shape control of the FGM plates under a temperature gradient by optimising (i) the voltage distribution for the open loop control, and (ii) the displacement control gain values for the closed loop feedback control. The effect of the constituent volume fractions of zirconia, through varying the volume fraction exponent n, on the optimal voltages and gain values has also been examined.     

碳纳米管增强型功能梯度板的屈曲预测

基于一种新的修正偶应力理论,建立了碳纳米管（CNTs）增强型功能梯度板（CNTs/FGP）的屈曲模型。基于最小势能原理和一阶剪切变形理论,推导了该种板模型的平衡微分方程和相应的边界条件,并以四边简支方板的屈曲问题为例,讨论了材料尺度参数、CNTs的体积分数及4种不同CNTs分布形式对CNTs/FGP临界屈曲载荷的影响。结果表明：采用本文模型预测的CNTs/FGP的临界屈曲载荷总是大于传统宏观理论的预测结果,两种理论结果间的差距随着板几何尺寸的减小而逐渐增大;CNTs体积分数的少量增加,即可使板的临界屈曲载荷有明显的提升;CNTs的不同分布形式对临界屈曲载荷有显著的影响,在工程设计中应予以关注。     

A design of laminated composite plates using graded orthotropic fiber-reinforced composite plies

A new lamination scheme is proposed through the design of a graded orthotropic fiber-reinforced composite ply for achieving continuous variations of material properties along the thickness direction of laminated composite plates. First, a micro-structure of graded unidirectional fiber-reinforced composite ply is designed and its effective graded elastic properties are estimated using finite element procedure. Next, the new lamination scheme is demonstrated through the conversion of a conventional laminated composite plate (CLCP) into a conventional-graded laminated composite plate (CGLCP) utilizing presently designed graded orthotropic composite ply. The suitability of this conversion/proposed lamination scheme is substantiated through the bending analysis of both the plates (CLCP and CGLCP).     

Three-dimensional thermomechanical buckling analysis for functionally graded composite plates

Three-dimensional thermomechanical buckling analysis is investigated for functionally graded composite structures that composed of ceramic, functionally graded material (FGM), and metal layers. Material properties are assumed to be temperature dependent, and in FGM layer, they are varied continuously in the thickness direction according to a simple power law distribution in terms of the ceramic and metal volume fractions. The finite element model is adopted by using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. Temperature at each node is obtained by solving the thermomechanical equations. For a time discretization, Crank–Nicolson method is used. In numerical results, the thermal buckling behavior of FGM composite structures due to FGM thickness ratios, volume fraction distributions, and system geometric parameters are analyzed.     

压电集成碳纳米管CNT增强功能梯度结构非线性建模与仿真

将碳纳米管(carbon nanotube, CNT)以梯度形式分布与基体材料结合,形成功能梯度(functionally graded,FG)结构.为了实现FG-CNT增强复合板在发生大变形时的准确计算,考虑四种典型的CNT分布形式,均匀分布、V型分布、O型分布和X型分布,建立基于Reissner-Mindlin板壳...     

Plastic limit analysis of cylindrically orthotropic circular plates

The plastic limit analysis of cylindrically orthotropic circular plates is developed using a piecewise linear orthotropic yield criterion. The yield criterion is a modification of an isotropic formulation that consists of a series of weighted piecewise linear components. The piecewise linear yield criterion enables an analytical solution for the plastic limit load of cylindrically orthotropic circular plates. Plastic limit analysis for both simply supported and clamped circular plates under uniformly distributed load are carried out. Parametric studies are conducted to investigate the sensitivity of the plastic limit loads to material orthotropy and influences of orthotropic ratio and chosen yield criteria on the plastic limit loads of the circular plates are discussed. It is found that the plastic limit loads of the orthotropic circular plates are affected significantly by the orthotropic ratio. Enhancement of the circumferential yield moment will increase dramatically the plastic limit load of the plates. Moment and velocity fields of the plates in plastic limit state are also derived and discussed. The results obtained from the present study are helpful in understanding the failure mechanism of orthotropic circular plates and is useful for design.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.