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主要基于三维弹性力学和状态空间法给出了四边简支各向同性矩形层合板自由振动和 强迫振动问题的精确解. 首先基于三维弹性力学建立了层合板的基本方程,利用状态空间法 解决了层合板的自由振动问题,然后根据Lagrange动力学方程求解了层合板受到横向冲击时 的强迫振动响应.     

六层非对称正交双稳态复合材料层合板的动态跳跃研究

本文研究了中心固支,四边自由的六层非对称正交双稳态复合材料层合板(0/0/0/90/90/90)的动态跳跃现象,并展示了双稳态复合材料层合板在动态跳跃前后和过程中的非线性振动.考虑冯卡门大变形,利用Reddy三阶剪切变形理论和哈密顿原理,建立了动力学与控制方程.利用动力学与控制方程,得到了李雅普诺夫指数图、庞加莱截面图...     

粘弹层合板的稳态振动和层间应力

应用混合分层理论和Ressiner混合变分原理，在板厚方向取二次位移插值函数和三次、四次横向应力插值函数推导出粘弹层合板的动力学方程，得出简支粘弹层合板稳态振动的解。不仅得出与三层弹性板精确的自振频率吻合良好的解，而且对于粘弹层合板，所计算的自振频率和结构损耗因子也与三维结果吻合较好。计算了自由阻尼层合板对应的低阶法向位移响应幅值和层问横向应力的幅值。结果表明，较高的层间横向正应力是低频稳态振动中引起粘弹层合板分层破坏的主要因素，采用适当模量和厚度的粘弹性材料将有效地降低粘弹层合板的层间横向正应力的幅值。     

具损伤压电智能层合板的动力分析

基于应变能等效原理、高阶剪切变形理论和Hamilton变分原理,考虑复合材料铺设层内的损伤效应,建立了具损伤压电智能层合板的运动控制方程,并运用Galerkin方法进行求解.数值算例中,讨论了损伤效应、厚跨比及压电层厚度与层合板总厚度之比对四边简支压电智能层合板自由振动频率的影响和外部控制电压对其动力响应的影响.     

含脱层层合板分叉屈曲的宏微观失效机制分析

根据板壳理论并考虑层合板脱层处的接触效应影响,推导了层合板的控制方程,借助量纲归一化分析建立了该问题的八次特征方程。通过对特征方程各参数进行定性分析,建立了求解八次方程根的解析表达式,发现了层合板的宏微观变形模态。利用MATLAB软件进行算例分析,获得了在不同脱层情况下层合板的屈曲模态参数和临界屈曲荷载。研究表明:层合板的屈曲模态包含整体一致的宏观变形和界面处的微观变形。研究结果与ABAQUS的有限元解进行了对比,结果表明:理论解与有限元分析的结果非常一致。论文从理论分析的角度研究了复合材料矩形层合板的屈曲问题,从理论上剖析了分层层合板的宏微观失效模态。     

有限长压电层合简支板自由振动的三维精确解

基于三维弹性理论和压电理论，导出了有限长矩形压电层合简支板的动力学方程及相应的边界条件，给出了一种求解压电层合板自由振动三维精确解的方法；分析了正、逆向压电效应对层合板振动频率的影响．本文所述的方法和结果对于求解其他三维动态问题，验证、比较其他简化模型、有限元计算结果以及工程应用都有指导意义．     

基于局部移动Kriging无网格方法的层合板自由振动分析

利用基于局部移动Kriging插值无网格法对层合板自由振动进行了数值分析,基于一阶剪切层合理论导出了层合板振动的控制方程和边界条件,进一步得到了自由振动的离散化特征方程。由于Kriging插值函数具有Kronecker delta函数性质,可以直接施加本质边界条件。通过本文给出的方法,对不同边界条件、不同跨厚比、不同材料参数和铺设角度的层合板的振动频率进行了计算,均得到满意结果。最后用该方法对层合板的铺设角度进行优化设计,得到了与已有文献完全一致的优化结果。数值结果充分表明了无网格Kriging方法分析层合板自由振动问题的有效性和高精确度。     

反对称铺设复合材料层合板非线性后屈曲分析

基于层合板壳理论,考虑反对称铺设层合板的拉弯耦合效应和后屈曲过程中的非线性几何变形,推导了由应力函数和挠度表示的复合材料层合板的后屈曲控制方程。引入无量纲参数对控制方程和边界条件进行无量纲化,以消除材料参数及几何尺寸对分析结果的影响。采用摄动法将无量纲的非线性控制方程及边界条件展开成一系列非齐次线性摄动方程组,分析各阶摄动方程的通解与特解的构造,并逐次求解,建立了反对称铺设复合材料层合板受单向均布压力作用的临界屈曲荷载及后屈曲平衡路径的理论解。进而运用ABAQUS软件对复合材料层合板在面内压缩载荷作用下的屈曲和后屈曲进行有限元分析,结果表明理论解与ABAQUS结果十分接近,验证了理论解的正确性。在此基础上进一步讨论了铺设角度、铺设层数和拉弯耦合效应等对层合板后屈曲性能的影响。研究发现层合板的屈曲载荷受铺设角度与层数的影响较为显著,而拉弯耦合效应使板的屈后强度大大降低。     

热环境下贴压电层的功能梯度材料板的自由振动和动力响应

基于Reddy高阶剪切变形理论和广义Kármán型方程,用双重Fourier级数展开法求得了热环境下带压电层的功能梯度复合材料混合层合板的自由振动及动力响应的解析解,分析中考虑了材料热物参数对温度变化的依赖性,讨论了环境温度和控制电压对固有频率及动力响应的影响。     

复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法

基于一阶剪切变形理论,提出了复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法。计算时在复合材料层合板中面上仅需要布置一系列的离散节点,并利用这些节点构建插值函数。在板中面上的局部多边形子域上,采用加权余量法建立复合材料层合板自由振动分析的离散化控制方程,并且这些子域可由Delaunay三角形方便创建。自然邻接点插值形函数具有Kronecker delta函数性质,因而无需经过特别处理就能准确地施加本质边界条件。对不同边界条件、不同跨厚比、不同材料参数和不同铺设角度的复合材料层合板,由本文提出的无网格自然邻接点Petrov-Galerkin法进行自由振动分析时均可得到满意的结果。数值算例结果表明,本文方法求解复合材料层合板的自由振动问题是行之有效的。     

矩形夹层板稳定性分析

本文在Reissner 理论基础上，应用功的互等定理推导了夹层矩形板稳定问题的基本解，在已推导出的夹层板基本解的基础上，利用功的互等法求解了两对边固定一边简支一边自由、两邻边简支另两邻边自由且角点支承、两邻边简支另两邻边自由且角点悬空三种不同边界条件下夹层板的稳定问题，给出了挠曲面方程及其对应的执行方程；进行了数值计算，并与有限元结果进行对照分析．结果表明：本文方法求解过程更简单，提供了一种求解夹层板稳定问题的新方法，计算结果对解决工程实际问题具有一定的参考价值．     

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.     

Analytical solution for buckling of VAT composite laminates with elastic restraints on two opposite edges

Meccanica - Based on classical laminated plate theory, an analytical solution for buckling of variable angle tow (VAT) composite laminates with elastic restraints on two opposite edges and simply...     

NONLINEAR DYNAMICS OF A FGM PLATE WITH TWO CLAMPED OPPOSITE EDGES AND TWO FREE EDGES

This paper deals with the nonlinear forced vibration of FGM rectangular plate with a boundary of two edges clamped opposite and the other two free. The plate is subjected to transversal and in-plane excitations. The present research treats the material properties of the FGM plates as temperature-dependent and graded continuously throughout the thickness direction, following the volume fraction of the constituent materials according to the power law. The temperature is assumed to be constant in the plane and varied only in the thickness direction of the plate. In the framework of geometrical nonlinearity the plate is modeled and the equations of motion are obtained on Hamilton＇s principle. With the help of Galerkin discretization, the nonlinear ordinary differential equations describing transverse vibration of the plate are proposed. By the numerical method, the nonlinear dynamical responses of the FGM plate with two clamped opposite and two free edges are analyzed.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

High order behavior of sandwich plates with free edges––edge effects

The edge effects of a sandwich plate with a “soft” core and free edges, i.e. the plate is supported only at the lower face-sheet, and the upper face-sheet and the core are free of stresses at their edges, using the high order approach (HSAPT), are presented. The two-dimensional analysis consists of a mathematical formulation that uses the classical thin plate theory for the face-sheets and a three-dimensional elasticity theory for the core. The governing equations and the required boundary conditions are derived explicitly through variational principals, yielding a system of eight partial differential equations. The non-homogeneous differential equations system is numerically solved using a modification of the extended Kantorovich method (MEKM). The model presented enables a two-dimensional solution of the stress and displacement fields when subjected to a general scheme of loads. It is applicable to any type of boundary conditions that can be applied separately on each face-sheet and on the core. A numerical study is presented, and it examines the behavior and the two-dimensional stress field of a sandwich plate with free edges, at the upper face-sheet and core, subjected to thermal and uniformly distributed loads, for various boundary conditions at the lower face-sheet. For completeness, the MEKM solution of the two-dimensional high order model is verified through comparison with a three-dimensional Finite Element model revealing good correlation. Furthermore, the problems involved in the construction of an appropriate three-dimensional FE model of a full scale sandwich plate that require large computer resources are discussed.The numerical study yields that the peeling (normal) stresses, which reach their maximum values at the edges of the sandwich plate, using a one-dimensional analysis, varies also in the transverse direction from a maximum value in the middle of the edge, descending towards the corners. Moreover, the nature of variation along the boundaries strongly depends on the type of loading and the transverse boundary conditions. The substantial variation of the stress field in the transverse direction clearly shows the necessity of a two-dimensional analysis and the inefficiencies of the one-dimensional model.     

Three-Dimensional Elasticity Solutions for Rectangular Orthotropic Plates

A state space approach for three-dimensional analysis of rectangular orthotropic elastic plates subjected to external loads on the top and bottom faces is developed. Through Hamiltonian variational formulation via Legendre’s transformation, the basic equations of elasticity are formulated into the state space framework in which the state equation exhibits Hamiltonian characteristics and the associated eigensystem possesses symplectic orthogonality. By means of separation of variables and eigenfunction expansion, three-dimensional elasticity solutions for orthotropic rectangular plates with two opposite edges simply supported and the other two arbitrary—which can be any combinations of simply-supported, clamped, and traction-free edges—are determined in a systematic way. The existing elasticity solution for the fully simply-supported plate is recovered. The through-thickness variations of the displacements and stresses are evaluated within the context.     

弹性地基上矩形薄板问题的Hamilton正则方程及解析解

利用辛算法求出弹性地基上矩形薄板问题的解析解,将弹性地基视为双参数弹性地基,直接从弹性矩形薄板的控制方程推导出了问题的Hamilton正则方程,为求出任意边界条件下问题的理论解奠定了基础,并且通过算例验证了文中所采用方法的正确性．     

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated. This paper is partially supported by the Deanship of Scientific Research at King AbdulAziz University (Grant no. 172/427).     

Axisymmetric bending for thick laminated circular plate under a concentrated load

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