The BEM applied to plate bending elastoplastic analysis using Reissner''s theory

An application of the boundary element method (BEM) to plate bending elastoplastic analysis is presented. Reissner's plate bending theory, which caters to thin and thick plates, is considered.

First, the governing equations are shown, in which bending plastic strains are allowed for. Thereafter, the integral equations are presented, including those for moments and shear resultants at internal points. The numerical implementation is carried out using the integral equations discretized in quadratic boundary elements and constant internal cells. An incremental-iterative method is employed to solve the elastoplastic equations.

Numerical examples are presented at the end of the work to illustrate the applicability of the formulation.     

Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory

ABSTRACT

This article investigates the nonlinear vibration of piezoelectric nanoplate with combined thermo-electric loads under various boundary conditions. The piezoelectric nanoplate model is developed by using the Mindlin plate theory and nonlocal theory. The von Karman type nonlinearity and nonlocal constitutive relationships are employed to derive governing equations through Hamilton's principle. The differential quadrature method is used to discretize the governing equations, which are then solved through a direct iterative method. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, external electric voltage, and temperature rise on the nonlinear vibration characteristics of piezoelectric nanoplates.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.     

Bending analysis of functionally graded sectorial plates using Levinson plate theory

Based on the Levinson plate theory (LPT) and the first-order shear deformation plate theory (FST), the bending analysis of functionally graded (FG) thick circular sector plates is presented. The LPT solutions of FG sectorial plates are first expressed in terms of the solutions of the classical plate theory (CPT) for homogeneous sectorial plates and then presented using a direct method. It is assumed that the non-homogeneous mechanical properties of plate, graded through the thickness, are described by a power function of the thickness coordinate. The results are given in closed-form solutions and verified with the known data in the literature.     

Transient and pseudo-transient analysis of Mindlin plates

A unified approach is presented for the static and transient dynamic linear and geometrically nonlinear analysis of Mindlin plates including initial imperfections. The effects of transverse shear deformation and rotatory inertia are automatically taken into account. A finite element idealization is adopted and the quadratic Lagrangian Mindlin plate element is used together with selective integration. Several numerical examples are presented and compared with results from other sources.     

Dynamic analysis of inelastic plates by the D/BEM

A direct boundary element method is developed for the dynamic analysis of thin inelastic flexural plates of arbitrary planform and boundary conditions. It employs the static fundamental solution of the associated elastic problem and involves not only boundary integrals but domain integrals as well. Thus boundary as well as interior elements are employed in the numerical solution. Time integration is accomplished by the explicit algorithm of the central difference predictor method. A viscoplastic constitutive theory with state variables is employed to model the material behaviour. Numerical results are also presented to illustrate quantitatively the proposed method of solution.     

Dynamic analysis of isotropic composite plates using a layerwise theory

The paper is concerned with small amplitude vibrations of moderately thick composite plates. Polygonal and simply supported plates of three asymmetrically arranged layers in perfect bond are considered. The homogeneous layers are assumed to consist of isotropic linear elastic materials. The Mindlin–Reissner kinematic assumptions are implemented separately to each layer, and as such model both the global and local elastic response of composite plates. The continuity of the transverse shear stress across the interfaces is prescribed according to Hooke's law. For plates composed of layers with a uniform ratio of mass density to shear modulus and a common Poisson's ratio the lateral deflection can be calculated independently from the in-plane displacements. It is shown that the lateral deflection is governed by a boundary value problem of the fourth order. Alternatively, by neglecting the longitudinal as well as rotary inertia the complex problem reduces to the simpler case of a homogenized shear-deformable plate with effective stiffness and corresponding set of boundary conditions. The theory is applied to statically and dynamically loaded rectangular simply supported composite plates with various dimensions and material properties.     

Transient dynamic and damping analysis of laminated anisotropic plates using a refined plate theory

A refined model is presented for the linear transient dynamic and damping analysis of laminated anisotropic composite plates. Experimental measurements of specific damping capacity of unidirectional composite beams are used to predict the specific damping capacity of laminated composite plates in various modes of vibration. A finite element idealization is adopted, and the quadratic Lagrangian element is used together with selective/reduced integration. A viscous damping approximation is then employed to calculate the damped transient response of laminated plates. The effects of transverse shear deformation, symmetry condition, boundary conditions, anisotropy, aspect ratio, fibre orientation and the lamination scheme on specific damping capacity and damped transient response are investigated. Realistic examples illustrate the importance of these parameters. The present results agree very closely with experimental results available in the literature and can serve as a benchmark for future comparison by other investigators.     

BEM based on discontinuous solutions in the theory of Kirchhoff plates on an elastic foundation

We examine the construction of discontinuous solutions for Kirchhoff plates on a generalized elastic foundation. By discontinuous solutions we mean solutions which, when crossing certain lines, have discontinuities of the first type. In the theory of Kirchhoff plates, there may be jumps of the transverse deflection, slope angle, bending moment and equivalent shear force. Initially we construct the solutions due to concentrated jumps. Using them as Green functions, we express discontinuous solutions that are the base for the indirect method of boundary integral equations.     

A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation

This communication discusses a 4-node plate bending element for linear elastic analysis which is obtained, as a special case, from a general nonlinear continuum mechanics based 4-node shell element formulation. The formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate (and shell) element.     

Further developments in transient and pseudo-transient analysis of Mindlin plates

Some recent developments are presented on a unified approach to the static and transient dynamic geometrically nonlinear analysis of Mindlin plates with initial imperfections.     

The frequency-temperature analysis equations of piezoelectric plates with Lee plate theory

Frequency-temperature analysis theory of crystal plates based on the incremental field theory (IFT) and the Mindlin plate theory have been widely used in the vibration analysis of crystal resonators subject to temperature change. As one of the two major plate theories for the resonator analysis, the Lee plate theory based on the trigonometric series expansion of displacements has been extensively used for both analytical and numerical analyses of quartz resonators, and efforts have also been made in correcting and perfecting the theory for much broader applications. In this paper, the earlier frequency-temperature analysis equations based on the IFT is further extended to the trigonometric series expansion in a systematic manner. By incorporating the frequency-temperature analysis into the Lee plate theory, the complete analysis of crystal resonators can be made in a consistent way. The thickness-shear and flexural vibration equations have been compared with the Mindlin plate theory to demonstrate the similarity and consistency.     

An alternative treatment of body forces in the BEM for thick plates resting on elastic foundations

In this paper, domain integrals due to body forces or uniform loading in the BEM for thick plates resting on elastic foundations are transformed to equivalent boundary integrals. Unlike, common techniques of the transformation, which are based on the Green second identity, the present formulation employs the Green first identity. The necessary particular solutions are derived and the kernels for the computation of stress resultants at the internal points are derived and given in explicit forms. The main advantages of the present formulation are the derived kernels are not singular and simple if compared to those of the Green second identity technique. A general technique for avoiding the appearance of jump terms in the transformed boundary integrals is presented. Three numerical examples are presented to demonstrate the accuracy and the validity of the new boundary integrals.     

分数阶微分双参数黏弹性地基矩形板动力响应

基于弹性地基Pasternak双参数模型,利用分数阶微分得到黏弹性地基双参数模型,并在此基础上建立采用分数阶微分Kelvin模型的双参数黏弹性地基上弹性和黏弹性矩形板在动荷载作用下的动力方程;利用Galerkin方法和分段处理的数值计算方法求解四边简支的弹性和黏弹性地基板的动力方程,通过自由振动算例验证该求解方法的正确性;并分析冲击动荷载作用下分数阶微分Kelvin模型的分数阶、粘滞系数、水平剪切系数和模量参数对位移响应的影响。结果表明：分数阶微分黏弹性模型可以描述不同黏弹性材料的力学行为;分数阶取值0.5前后,矩形板位移响应值出现了不同的衰减发展形态;粘滞系数、水平剪切系数和模量系数取值越大,位移响应衰减速度越快。     

Vibration analysis of orthotropic rectangular plates on elastic foundations

The vibration responses of orthotropic plates on nonlinear elastic foundations are numerically modeled using the differential quadrature method. The differential quadrature technique is utilized to transform partial differential equations into a discrete eigenvalue problem. Numerical results and those from literature closely correspond to each other. Numerical results demonstrate that elastically restrained stiffness, plate aspect ratio and foundation stiffness significantly impact the dynamic behavior of orthotropic plates.     

采用高斯展开法分析组合式开口矩形Mindlin板的弯曲自振特性EI北大核心CSCD

基于Mindlin板理论,将高斯展开法引入到组合式开口矩形板弯曲振动问题研究中。选取高斯小波函数作为位移形函数,其本身的局域化特性能够准确捕捉到局部开口处的特征,提高计算效率;定义了高斯函数的伸缩因子和平移因子,通过伸缩和平移变换生成一系列用于拟合开口板位移场的基函数,增大伸缩因子取值使得形函数具有更高的分辨率,进而能更精确地模拟更高频的振动场;以能量法为基本框架,建立了开口板的拉格朗日能量泛函,并引入人工弹簧模型模拟各种边界条件,将边界条件以弹性势能的方式附加到开口板的能量泛函之中。通过与有限元软件的计算结果对比,验证了该方法的准确性与适用性。     

Mathematical analysis and numerical study to free vibrations of annular plates using BIEM and BEM

In this paper, the spurious eigenequations for annular plate eigenproblems by using BIEM and BEM are studied in the continuous and discrete systems. Since any two boundary integral equations in the plate formulation (4 equations) can be chosen, 6 (C) options can be considered instead of only two approaches (single‐layer and double‐layer methods, or singular and hypersingular equations) which are adopted for the eigenproblems of the membrane and acoustic problems. The occurring mechanism of the spurious eigenequation for annular plates in the complex‐valued formulations is studied analytically. For the continuous system, degenerate kernels for the fundamental solution and the Fourier series expansion for the circular boundary density are employed to derive the true and spurious eigenequations analytically. For the discrete system, the degenerate kernels for the fundamental solution and circulants resulting from the circular boundary are employed to determine the true and spurious eigenequations. True eigenequation depends on the specified boundary condition while spurious eigenequation is embedded in each formulation. It is found that the spurious eigenvalue for the annular plate is the true eigenvalue of the associated interior problem with an inner radius of the annular domain. Also, we provide three methods (SVD updating technique, Burton and Miller method and CHIEF method) to suppress the occurrence of the spurious eigenvalues. Several examples were demonstrated to check the validity of the formulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary element analysis of a plate on elastic foundations

The objective of this study is to examine the applicability of the boundary element method to analysing a plate on elastic foundation. The fundamental solution of the problem is presented as a Fourier-Bessel integral. For the computation of the values of the fundamental solution an algorithm was developed in which the Fourier-Bessel integral was decomposed into an alternative convergent sequence. Equations based on the direct and indirect boundary element method were derived for a plate situated on a one- or two-parametric elastic foundation. According to the theory presented, computer programs based on the direct and indirect boundary element method were developed. These programs can be used for examining the behaviour of a smooth-boundary plate on a one- or two-parametric elastic foundation. The computer programs were tested by several examples. The results obtained by using a small number of boundary elements compared favourably to the results obtained by a fine finite element mesh. The study shows that the boundary element method is applicable to the analysis of a plate on elastic foundation.     

Consideration of stiffness and mass effects of relatively thicker electrodes with Mindlin plate theory

Mindlin plate theory has been widely used in the high-frequency vibrations of piezoelectric crystal plates with emphasis on its applications in crystal resonator analysis and design. The plate equations were derived without considering the effect of electrodes from the beginning. But continuing efforts have been made to include the mechanical effect, or the mass loading, through the consideration of the mass ratio of the electrodes and crystal blank. Such a consideration has been effective for relatively thin electrodes before, but the ever-increasing mass ratio has been pressing further improvement to take into account relatively thicker electrodes. To extend Mindlin plate equations for these applications, we derive the plate equations systematically with the approximation of displacements in electrodes with those in the crystal blank. As a result, both mass and stiffness effects of electrodes are considered through ratios of the thickness, density, and elastic constants of the electrodes to those of the crystal blank, respectively, and the plate equations are modified accordingly. A practical design of the electrodes and crystal blank are analyzed to demonstrate the necessity of such modifications to Mindlin plate equations.