具有自由边的复合材料层合板的解析解

从三维弹性力学基本方程出发,通过假设自由边的边界位移函数,建立了正交异性层合板的状态方程,给出了对边自由,对边简支矩形板的解析解.此解满足层合板的基本方程和层间连续条件.用本文的方法比较容易处理层合板的自由边.算例表明,数值结果具有较高的精度.     

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．     

功能梯度材料矩形板的三维静动态响应分析

基于线弹性理论的基本方程,选用3个位移分量和3个应力分量作为状态变量,利用状态空间法建立了功能梯度矩形板的三维状态方程. 考虑四边简支的边界条件,采用打靶法数值求解了材料常数沿板厚按幂率变化的弯曲问题和自由振动问题,为求解功能梯度材料三维弹性响应提供了一种方法. 并且给出了功能梯度材料三维矩形板的静动态响应受组分材料分布以及板厚长比变化的影响规律.      

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。     

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。     

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。     

四边简支各向同性矩形厚板的插值矩阵法解

针对强厚度矩形板四边简支情况,论文根据状态变量法思想,基于三维弹性理论基本方程,以3个位移分量及3个应力分量按双三角级数展开,将三维弹性力学控制方程转化为常微分方程边值问题.尽管一些各向异性弹性矩形厚板早已由状态空间法获得分析解,可是各向同性厚板的分析解至今难以获得,因为状态空间解法中特征方程有重根问题而不易于收敛.论文提出采用插值矩阵法直接对常微分方程进行求解,获得各向同性矩形厚板在四边简支边界条件下三维理论的位移和应力解,并与有限元精细结果进行比较,证明了本文解的准确性.     

湿热环境对挖补修理后复合材料层合板振动特性的影响

通过对湿热环境下阶梯挖补胶接修理后碳纤维/双马树脂(T300/QY8911)层合板振动特性的分析,探究了挖补修理后碳纤维/双马树脂层合板在不同湿热效应下的振动特性。基于Mindlin-Reissner板理论与最小势能变分原理,在考虑湿热效应前提下,推导了正交各向异性复合材料层合板的本构关系及四边简支条件下的自由振动控制方程;其次,利用ABAQUS软件建立了三维黏弹性有限元挖补修理模型,应用Fortran语言将湿热耦合下的应力-应变本构关系编入UMAT二次开发文件中,采用三维八节点五自由度等参单元进行了离散。通过与参考文献对照,验证了胶接修理后T300/QY8911层合板有限元模型和编译的湿热环境下UMAT子程序的合理性与有效性。算例结果表明:湿热环境对挖补修理后层合板固有频率的影响比对完整层合板严重;温度对材料性能的影响大于湿度;相比于材料性能变化对振动特性的影响,湿热残余应力对胶接修理后层合板振动特性的影响占主导;参考环境下,胶接修理后层合板的固有频率增大;湿应力对层合板固有频率的影响大于热应力。     

功能梯度矩形板的三维弹性分析

将功能梯度三维矩形板的位移变量按双三角级数展开，以弹性力学的平衡方程为基础．导出位移形式的平衡方程。引入状态空间方法，以三个位移分量及位移分量的一阶导数为状态变量，建立状态方程。考虑四边简支的边界条件，由状态方程得到了功能梯度三维矩形板的静力弯曲问题和自由振动问题的精确解。由给出的均匀矩形板自由振动问题的计算结果表明．与已有的理论解以及有限元方法的计算结果相吻合。假设功能梯度三维矩形板的材料常数沿板的厚度方向按照指数函数的规律变化．进一步给出了功能梯度三维矩形板的自由振动问题和静力弯曲问题的算例分析，并讨论了材料性质的梯度变化对板的动力响应和静力响应的影响。     

含压电层的功能梯度柱壳的自由振动分析

基于三维弹性理论,导出了带有压电层的圆柱形梯度壳的动力学方程以及相应的边界条件,用幂级数展开法得到了求解该圆柱形梯度壳自由振动的三维精确公式．通过实例模型求解了该壳体的自由振动的固有频率;分析了不同电学边界条件对壳体的振动频率的影响。结果可评估各种近似理论解和数值解的正确性。     

Three dimensional elasticity solution for vibration problem of thick plate

I.Introduction'TheclassicalplatetheoryofvibrationisbasedontheKirchhoffhypotllcsis.andtheeffectoftransversesheardeformationandrotatoryinertiaareneglected.111[lltllcfreevibratiollfrequenciesofthinplates,allkindsofboundaryconditionshavebeensolved.Fortheanalysisofthevibrationalproblemsofthickplate,thetheoryofReissuer,MindlinandHenkyisused.Accordingtothedegreesoftheeffectsoftransverseshearstress,transversenormalstressandrotatoryinertia,somedynamicproblemsinEngineeringhavebeensolvedl:l.Regardless…     

厚圆板轴对称振动的弹性力学解

本文以轴对称三维弹性力学基本方程为基础,导出厚圆板强迫振动的状态方程式。利用Ｍａｃｌａｕｒｉｎ级数和Ｓｙｌｖｅｓｔｅｒ定理,厚圆板的位移和应力可以用中面位移和应力的微分算子表示。通过载荷分解和圆板表面条件,可以得到厚圆板在对称载荷与反对称载荷作用下的振动控制方程。求解了厚圆板在周边固支和简支条件下的对称与反对称的自由振动问题。通过数值计算得到了这两类自由振动的固有频率。本文的方法适用于求解厚圆板在     

Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell,cylindrical shell,and annular plate using theory of elasticity and DQM

Abstract

In this paper, three-dimensional static and free vibration analysis of functionally graded graphene platelets-reinforced composite (FG-GPLRC) truncated conical shells, cylindrical shells and annular plates with various boundary conditions is carried out within the framework of elasticity theory. The main contribution of the present work is that formulation for free vibration and bending behavior of the FG-GPLRC truncated conical shell based on theory of elasticity has not yet been reported. Additionally, formulation and solution for cylindrical shell and annular plate are derived by changing the semi vertex angle in formulation and solution of FG-GPLRC truncated conical shell. A semi-analytical solution is proposed base on employing differential quadrature method (DQM) together with state-space technique. Validity of current approach is assessed by comparing its numerical results with those available in the literature. An especial attention is drawn to the role of GPLs weight fraction, patterns of GPLs distribution through the thickness direction, geometrical parameters such as semi-vertex angle, length to mid-radius ratio on natural frequencies and bending characteristics. Numerical results reveal that desirable static and free vibration response (such as lower radial deflection and higher natural frequencies) can be achieved by locating more square shaped GPLs near inner and outer surfaces.     

双周期加筋微穿孔板的振动响应及声透射的研究

本文主要研究了水下无穷大双周期加筋微穿孔薄板,在平面声波斜入射下的振动响应和声透射,并提出了一种半解析半数值的计算方法。利用微穿孔板的声阻抗以及薄板表面的振速边界条件,建立了加筋穿孔薄板的振动方程,并根据傅立叶变换及空间波数法将振动位移表达为波数分量的迭加形式。采用数值计算的方法对波数分量进行求解并通过傅里叶逆变换,最终得到了双周期加筋穿孔薄板的振动响应及透射系数。通过与Takahashi穿孔板声压结果的对比,证明了本方法的正确性。在算例中,分析了加强筋及穿孔率对薄板结构的振动和声透射的影响。     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

Analysis of plate in second strain gradient elasticity

The bending analysis of a thin rectangular plate is carried out in the framework of the second gradient elasticity. In contrast to the classical plate theory, the gradient elasticity can capture the size effects by introducing internal length. In second gradient elasticity model, two internal lengths are present, and the potential energy function is assumed to be quadratic function in terms of strain, first- and second-order gradient strain. Second gradient theory captures the size effects of a structure with high strain gradients more effectively rather than first strain gradient elasticity. Adopting the Kirchhoff’s theory of plate, the plane stress dimension reduction is applied to the stress field, and the governing equation and possible boundary conditions are derived in a variational approach. The governing partial differential equation can be simplified to the first gradient or classical elasticity by setting first or both internal lengths equal to zero, respectively. The clamped and simply supported boundary conditions are derived from the variational equations. As an example, static, stability and free vibration analyses of a simply supported rectangular plate are presented analytically.     

粘弹层合板的自由振动和横向应力

Based on Reddy＇s layerwise theory, the governing equations for dynamic response of viscoelastic laminated plate are derived by using the quadratic interpolation function for displacement in the direction of plate thickness. Vibration frequencies and loss factors are calculated for free vibration of simply supported viscoelastic sandwich plate, showing good agreement with the results in the literature. Harmonious transverse stresses can be obtained. The results show that the transverse shear stresses are the main factor to the delamination of viscoelastic laminated plate in lower-frequency free vibration, and the transverse normal stress is the main one in higher-frequency free vibration. Relationship between the modulus of viscoelastic materials and transverse stress is analyzed. Ratio between the transverse stress＇s maximum value and the in-plane stress＇s maximum-value is obtained. The results show that the proposed method, and the adopted equations and programs are reliable.     

Three-dimensional vibrations of annular thick plates with linearly varying thickness

The free vibration of annular thick plates with linearly varying thickness along the radial direction is studied, based on the linear, small strain, three-dimensional (3-D) elasticity theory. Various boundary conditions, symmetrically and asymmetrically linear variations of upper and lower surfaces are considered in the analysis. The well-known Ritz method is used to derive the eigen-value equation. The trigonometric functions in the circumferential direction, the Chebyshev polynomials in the thickness direction, and the Chebyshev polynomials multiplied by the boundary functions in the radial direction are chosen as the trial functions. The present analysis includes full vibration modes, e.g., flexural, thickness-shear, extensive, and torsional. The first eight frequency parameters accurate to at least four significant figures for five vibration categories are obtained. Comparisons of present results for plates having symmetrically linearly varying thickness are made with others based on 2-D classical thin plate theory, 2-D moderate thickness plate theory, and 3-D elasticity theory. The first 35 natural frequencies for plates with asymmetrically linearly varying thickness are compared to the finite element solutions; excellent agreement has been achieved. The asymmetry effect of upper and lower surface variations on the frequency parameters of annular plates is discussed in detail. The first four modes of axisymmetric vibration for completely free circular plates with symmetrically and asymmetrically linearly varying thickness are plotted. The present results for 3-D vibration of annular plates with linearly varying thickness can be taken as benchmark data for validating results from various plate theories and numerical methods.     

A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate

Based on Mindlin's plate theory, free vibration analysis of moderately thick shear deformable annular functionally graded plate coupled with piezoelectric layers is presented in this paper. A consistent formulation that satisfies the Maxwell static electricity equation is presented so that the full coupling effect of the piezoelectric layer on the dynamic characteristics of the annular FGM plate can be estimated based on the free vibration results. The differential equations of motion are solved analytically for various boundary conditions of the plate through the transformation of variable method. The applicability of the proposed model is analyzed by studying the effect of varying the gradient index of FGM plate on the free vibration characteristics of the structure. For some specific cases, obtained results were cross checked with those existing literatures and furthermore, verified by those obtained from three-dimensional finite element (3D FE) analyses.     

弹性半空间地基上预应力中厚矩形板的横向振动

基于Reissner-Mindlin一阶剪切变形理论,讨论在预加面内机械荷载作用下,弹性半空间地基上四边自由中厚矩形板的横向振动问题。把地基看作三维弹性半空间体,考虑地基变形的衰减。用一组数学上完备的二元多项式作为位形函数,采用pb-2 Rayleigh-Ritz法求得四边自由中厚矩形板的自振频率和在横向简谐荷载作用下的动力响应。讨论了板的长宽比、宽厚比及弹性地基和板的相对刚度对板的自振频率的影响。