层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

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.     

弹性矩形板问题的Hamilton正则方程

为了采用辛算法求出弹性矩形板问题的解析解，中直接从弹性矩形板的控制方程出发推导了弹性矩形板，其中包括弹性矩形薄板和厚板问题以及弹性地基上矩形薄板和厚板问题的Hamilton正则方程，为利用辛几何方法求出任意边界条件下这类问题的理论解奠定了基础．     

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.     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K1 与K2的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material(FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-ofvariables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.     

梯度弹性基础上正交异性薄板的弯曲

基于能量法和变分原理,采用双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板在分布载荷作用下的弯曲问题.首先,根据能量法与变分原理,给出了梯度弹性基础上正交异性薄板的弯曲微分平衡方程,并得到了梯度弹性基础刚度系数K1与K2的计算表达式;进而,假设z向正应力在厚度方向上均匀分布,推导了弹性基础z向位移衰减函数(ψ)(z)的计算式.在算例中,通过将梯度弹性基础退化为均质基础,并与Vlazov模型对比,证明了论文理论的正确性;最后,求解了弹性模量呈幂律分布的梯度基础上薄板的挠度分布,分析了基础上下表层材料弹性模量比λ与体积分数指数n对薄板挠度分布的影响.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

横观各向同性弹性半空间地基上四边自由各向异性矩形薄板弯曲解析解

选用更具广泛性的横观各向同性弹性半空间地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解．将异性薄板的弯曲控制方程,与基于横观各向同性弹性半空间地基位移解建立的板与地基变形协调方程相结合,先按对称性分解,然后用三角级数法,得出横观各向同性弹性半空间地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及内力的解析表达式．该解析解克服了数值法的弊端,取消了对地基反力的假设,板的内力及地基反力求解更切实际．算例结果与文献结果吻合良好,证明本文方法的可行性．     

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.     

Global bifurcations and homoclinic trees in motion of a thin rectangular plate on a nonlinear elastic foundation

The global bifurcations and chaotic dynamics of a thin rectangular plate on a nonlinear elastic foundation subjected to a harmonic excitation are investigated. On the basis of the amplitude and phase modulation equations derived by the method of multiple scales, a near integrable two-degree-of-freedom Hamiltonian system is obtained by a transformation. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the thin rectangular plate. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, which implies that chaotic motions may occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures and numerical evidence of chaotic motions.     

A free rectangular plate on the two-parameter elastic foundation

This paper provides a rigorous solution of a free rectangular plate on the V.Z. Vlazov two-parameter elastic foundation by the method of superposition[1]. In this paper we derive basic solutions under the various boundary conditions. To superpose these basic solutions the most generally rigorous solution of a free rectangular plate on the two-parameter elastic foundation can be obtained. The solution strictly satisfies the differential equation of a plate on the two-parameter elastic model foundation, the boundary conditions of the free edges and the free corner conditions. Some numerical examples are presented The calculated results show that when the plane dimension of plate is given and the ratio between the laver depth and the plate thick is equal to 15, the two-parameter elastic model is near the Winkler’s. It shows that the Winkler model can be applied to the thinner layer.     

Free vibration analysis of general stepped circular plates with internal elastic ring support resting on Winkler foundation by green function method

Natural frequencies are important dynamic characteristics of a structure. Therefore, the exact solution pertaining to free vibration of stepped circular plate elastically restrained against rotation, translation, and internal elastic ring support resting on an arbitrary variable elastic foundation using Green Function is presented in this paper. Thus, an accurate and direct modeling technique is introduced for modeling stepped circular plate on an arbitrary variable elastic foundation with arbitrary boundary conditions and internal elastic ring support. The effect of the translational along with rotational support flexibilities, as well as, the elastic coefficient of Winkler foundation and other parameters are assessed. Finally, some numerical examples are shown in order to present the efficiency and simplicity of the Green Function in the new formulation.     

弹性地基上含孔隙功能梯度材料板的自由振动和动力响应

为研究弹性地基上含孔隙的材料特性沿厚度呈Sigmoid函数变化的功能梯度材料(S-FGM)板的振动特性,本文基于改进的Voigt模型,分别建立了孔隙为均匀分布和非均匀分布两种类型的功能梯度材料的物性参数模型.根据复合材料薄板理论导出了弹性地基上含孔隙的功能梯度材料板的运动方程,用伽辽金法寻求四边简支边界条件下板自由振动...     

基于厚板理论分析深水域中弹性浮板的水波响应

基于线性水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf 方法,研究了不同水深水面上弹性浮板在不同入射波数水波作用下的动力学响应问题。首先推导了无限深水域中弹性浮板水波响应的解析解,并将本文分析计算结果与采用其他方法(经典薄板理论)得到的计算结果进行了对比和分析;其次,采用本文方法研究了大型浮板在三种入射波数的水波作用下动弯矩幅值的分布情况;最后,根据其他文献的方法计算了不同水深(有限水深)情况下浮板的动响应,并与本文的计算结果进行了对比分析。     

Bending of annular sectorial Mindlin Plates using Kirchhoff results

This study is concerned with the elastic bending problem of a class of annular sectorial plates whose radial edges are simply supported. Exact bending relationships between the Mindlin plate results and the corresponding Kirchhoff plate solutions have been derived based on the concept of load equivalence. These bending relationships facilitate the deduction of thick (Mindlin) plate results from the corresponding classical thin (Kirchhoff) plate solutions, thus bypassing the need to solve the more complicated governing equations of thick plates. The correctness of the relationships is established by solving the bending problem of annular sectorial plates under a uniformly distributed load and comparing the results with existing thick plate solutions.     

A non-classical Kirchhoff plate model incorporating microstructure,surface energy and foundation effects

A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.     

Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory

Analytical solutions for bending, buckling, and vibration of micro-sized plates on elastic medium using the modified couple stress theory are presented. The governing equations for bending, buckling and vibration are obtained via Hamilton’s principles in conjunctions with the modified couple stress and Kirchhoff plate theories. The surrounding elastic medium is modeled as the Winkler elastic foundation. Navier’s method is being employed and analytical solutions for the bending, buckling and free vibration problems are obtained. Influences of the elastic medium and the length scale parameter on the bending, buckling, and vibration properties are discussed.     

THE RELATION OF VON KRMN EQUATION FOR ELASTIC LARGE DEFLECTION PROBLEM AND SCHRDINGER EQUATION FOR QUANTUM EIGENALUES PROBLEM

In this paper the solutions of von Karman for elastic large deflection problem areclassified as the several solutions of Schr(?)dinger equation for quantum eigenvaluesproblem,and we present the transfrom for elastic large deflection problem from non-linearequation into linear equation.Thus,we create favourable conditions of the adoption ofconverse scattering methnd and B(?)cklund transformation.We also discuss the largedeflection problem of long and narrow plate.We can study the non-linear transition of elastic thin plate by furnished method fromthis paper.