On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid

In this paper, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. The mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. Based on the recently proposed differential equation governing the dynamical responses of the MEE rectangular plates, a fluid-structure interaction model is established and analyzed. First of all, the added virtual mass incremental (AVMI) factor of the system is calculated by using the proposed approach and the added virtual mass can then be obtained, furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. It is noted that the natural frequencies based on the proposed method are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid.     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

中厚板的弹性屈曲和后屈曲

本文采用Reissner假定考虑横向剪切变形的影响,导出弹性矩形板大挠度方程.本文讨论考虑横向剪切变形的矩形板的弹性屈曲和后屈曲.采用文[8]提供的摄动方法,给出了完善和非完善中厚板的后屈曲平衡路径,并与经典薄板理论结果进行了比较.     

Static plane-strain deformation of transversely isotropic magneto-electro-elastic and layered cylinders to general surface loads

Anisotropic and layered cylinders are important composite structures; however, their system of governing equations is usually solved numerically due to the complicated geometry and material anisotropy involved. In this paper, we analytically solve the plane-strain equations for general static deformation of a cylindrically anisotropic, layered magneto-electro-elastic (MEE) cylinder. We assume that the layers are perfectly bonded at the interfaces. We solve the equations through separation of variables and eigenfunction expansion. Results for each mode shape (2π periodic) are solved independently. Because the eigenspace of the mode shapes is the set of all continuous functions on the interval, any continuous loading can be applied and the corresponding solution can be found analytically through superposition of the mode-shape results. To check our formulation, we consider a cylinder with two isotropic-elastic layers under simple radial loading and reproduce the known, exact results. Then, we compare our formulation to an FEA solution for a layered piezo-electric (PE) cylinder. Finally, we apply a radial stress to three comparable MEE cylinders (one uniform MEE cylinder and two layered cylinders made of alternating piezo-electric (PE) and piezo-magnetic (PM) materials). Deformation and stress amplitudes are plotted for the first six mode shapes of each cylinder as benchmarks for further reference.     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Accurate bending analysis of rectangular plates with two adjacent edges free and the others clamped or simply supported based on new symplectic approach

The symplectic geometry approach is introduced for accurate bending analysis of rectangular thin plates with two adjacent edges free and the others clamped or simply supported. The basic equations for rectangular plates are first transferred into Hamilton canonical equations. Using the symplectic approach, the analytic solution of rectangular thin plate with two adjacent edges simply supported and the others slidingly supported is derived. Accurate bending solutions of title problems are then obtained using the superposition method. The approach used in this paper eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results are presented to demonstrate the validity and efficiency of the approach as compared with those reported in other literatures.     

Double‐mode modeling of nonlinear flexural vibration analysis for a symmetric rectangular honeycomb sandwich thin panel by the homotopy analysis method

Nonlinear flexural vibration of a symmetric rectangular honeycomb sandwich thin panel with simply supported along all four edges is studied in this paper. The nonlinear governing equations of the symmetric rectangular honeycomb sandwich panel subjected to transverse excitations are simplified to a set of two ordinary differential equations by the Galerkin method. Based on the homotopy analysis method, the average equations of the primary resonance and harmonic resonance are obtained. The influence of structural parameters, the transverse exciting force amplitude, and transverse damping to the symmetric rectangular honeycomb sandwich panel are discussed by using the analytic approximation method. Compared with the results obtained by single‐mode modeling technique, the results obtained by double‐mode modeling technique change the softening and hardening nonlinear characteristics when Ω ≈ ω₁, ω₁/3, and ω₂/3.     

厚板在集中荷载作用下的弯曲

本文根据[1]中提出的简化理论,利用两变元的δ-函数的性质[2]和级数解法,处理了在集中荷载作用下两对边简支,另两对边为任意的矩形厚板的弯曲问题.考虑了横向剪力对于弯曲变形的影响.当板的厚度h很小时,忽略公式中所有h2以上的项,则所得的结果与薄板弯曲问题的相应解一致[3].在本文的最后,我们还得到了在任意线分布荷载作用下相应问题的解.     

Existence and uniqueness of weak solutions for a thin plate with elastic boundary conditions

Robin-type problems are studied for thin elastic plates with transverse shear deformation. These problems are reduced to analogous ones for the corresponding homogeneous equilibrium equation, whose solutions are then represented as single and double layer potentials. The unique solvability of the systems of boundary integral equations yielded by this procedure is discussed in Sobolev spaces.     

任意载荷作用下矩形弹性薄板横向弯曲的一个高精度近似解*

李兹法是近似求解弹性薄板横向弯曲的一种广泛使用的有效方法,其精度完全取决于基函数的选择.本文根据矩形薄板横向弯曲的特点,将基函数选择为正弦三角级数与多项式函数的叠加,不但公式简单易程序化,而且有着很高的精度.本文最后给出了两个算例,并与经典结果进行了比较.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.     

点简支正交各向异性矩形薄板弯曲的精确解

本文利用迭加原理,给出了点简支正交各向异性短形薄板弯曲问题的封闭的级数式解答.简支点的位置和横向载荷的分布均可任意.用本文的级数解给出的算例与以往的数值解是十分一致的.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

中厚板热后屈曲分析

依据Reissner-Mindlin板理论考虑转动惯量和横向剪切变形影响,本文给出中厚板在(1)均布和非均布(线性)热荷载作用下;(2)单向压缩和均布热荷载共同作用下的后屈曲分析。采用摄动法导出完善和非完善中厚板的热屈曲载荷和热后屈曲平衡路径,并与经典薄板理论结果进行了比较。     

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich-Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two-dimensional orthotropic thick plate theories with stress or mixed edge data.