弹性地基中含孔隙的功能梯度圆锥壳的振动分析

为研究弹性地基中多孔功能梯度材料圆锥壳的振动特性,基于经典薄壳理论建立了弹性地基中含均匀和非均匀分布孔隙的功能梯度材料圆锥薄壳的振动方程,并用伽辽金法求得了自由振动和动力响应的解。通过参数分析讨论了孔隙、弹性地基参数、半锥角等因素对功能梯度圆锥壳自由振动和动力响应的影响。结果表明,弹性地基的压缩和剪切刚度的增大提高了圆锥壳的振动频率而显著减小了动力响应;当半锥角增大时,圆锥壳的动力响应显著增大。与非均匀分布孔隙壳体相比,均匀分布孔隙壳体的自振频率和动力响应随孔隙率的变化更为敏感。     

黏弹性地基上功能梯度材料板的振动分析

为研究黏弹性地基上功能梯度材料板的自由和强迫振动特性,基于Reddy高阶剪切变形理论以及由Shen导得的广义Karman型方程,用双重Fourier级数法推导了三参数黏弹性地基上四边简支功能梯度材料板自由振动和动力响应的解析解,计算了各模态自振频率和半波冲击载荷作用下的动力响应,讨论了材料组分指数、黏弹性地基参数、边厚比等因素对自由振动和动力响应的影响.结果表明,黏弹性地基的剪切和压缩刚度显著提升了功能梯度材料板的振动频率,减小了动力响应;另外,地基的黏性对振动频率和动力响应也有一定的影响.     

粘弹性地基上含孔隙的石墨烯增强功能梯度板的振动

为分析粘弹性地基上含孔隙的石墨烯增强功能梯度板的自由和强迫振动特性,基于三参数粘弹性地基模型及复合材料薄板理论,建立了粘弹性地基上含孔隙石墨烯增强功能梯度板的运动方程,用伽辽金法求解其固有频率和动力响应,并通过数值算例分析了粘弹性地基参数、孔隙率、孔隙类型及石墨烯纳米片分布模式、含量等因素对自由振动和动力响应的影响．结果表明,固有频率随着孔隙率的增大非单调变化,孔隙率对固有频率的影响随着地基参数、孔隙类型的不同而不同．另外,在三种孔隙类型中,上下表面层含有最少孔隙数量的板的动挠度最小,且其动挠度随着孔隙率的增大而微弱提高．     

弹性地基上多孔功能梯度材料矩形板的自由振动与临界屈曲载荷分析

多孔功能梯度材料(FGM)构件的特性与孔隙率和孔隙分布形式有密切关系。本文基于经典板理论,考虑不同孔隙分布形式时修正的混合率模型,研究Winkler弹性地基上四边受压多孔FGM矩形板的自由振动与临界屈曲载荷特性。首先利用Hamilton原理和物理中面的定义推导Winkler弹性地基上四边受压多孔FGM矩形板自由振动的控制微分方程并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程和边界条件进行变换,得到计算无量纲固有频率和临界屈曲载荷的代数特征方程。将问题退化为孔隙率为零时的FGM矩形板并与已有文献进行对比以验证其有效性。最后计算并分析了梯度指数、孔隙率、地基刚度系数、长宽比、四边受压载荷及边界条件对多孔FGM矩形板无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

梯度波阻板的地基振动控制研究

波阻板(wave impeding block,WIB)隔振体系是一种有效的振动污染治理措施,虽逐渐被应用在工程实际中,但以往的研究多集中于单相固体均质材料的情形,而对材料特性沿空间连续变化的非均匀固体材料的波阻板隔振性能的研究相对较少.基于功能梯度材料(functionally graded material,FGM)特点,本文提出了以功能梯度波阻板作为隔振屏障的一类新型的地基振动控制体系.考虑在弹性地基内部设置梯度波阻板,基于线弹性理论,利用傅里叶积分变换,根据Helmholtz矢量分解原理,建立了弹性地基在动载荷作用下的回传射线矩阵法(reverberation ray matrix method,RRMM)计算列式.假设梯度波阻板的物理力学性质沿深度方向按幂函数连续变化,采用数值傅里叶逆变换获得了弹性地基的位移和应力等物理量的数值解.通过数值算例,与单相固体均质波阻板进行了对比,并分析讨论了梯度波阻板的材料梯度因子、埋深以及梯度波阻板厚度等物理力学参数对地基隔振性能的影响规律.结果表明,梯度波阻板能有效降低振动的振幅,与单相固体均质波阻板相比,梯度波阻板具有更好的减振隔振效果.地基的位移幅值和应力幅值随着梯度因子的增大而减小.梯度波阻板的隔振效果随着波阻板厚度的增大而提高,而随着梯度波阻板埋深的增大而降低.     

ANALYSIS OF GROUND VIBRATION CONTROL BY GRADED WAVE IMPEDING BLOCK

波阻板（wave impeding block,WIB）隔振体系是一种有效的振动污染治理措施,虽逐渐被应用在工程实际中,但以往的研究多集中于单相固体均质材料的情形,而对材料特性沿空间连续变化的非均匀固体材料的波阻板隔振性能的研究相对较少.基于功能梯度材料（functionally graded material,FGM）特点,本文提出了以功能梯度波阻板作为隔振屏障的一类新型的地基振动控制体系.考虑在弹性地基内部设置梯度波阻板,基于线弹性理论,利用傅里叶积分变换,根据Helmholtz矢量分解原理,建立了弹性地基在动载荷作用下的回传射线矩阵法（reverberation ray matrix method,RRMM）计算列式.假设梯度波阻板的物理力学性质沿深度方向按幂函数连续变化,采用数值傅里叶逆变换获得了弹性地基的位移和应力等物理量的数值解.通过数值算例,与单相固体均质波阻板进行了对比,并分析讨论了梯度波阻板的材料梯度因子、埋深以及梯度波阻板厚度等物理力学参数对地基隔振性能的影响规律.结果表明,梯度波阻板能有效降低振动的振幅,与单相固体均质波阻板相比,梯度波阻板具有更好的减振隔振效果.地基的位移幅值和应力幅值随着梯度因子的增大而减小.梯度波阻板的隔振效果随着波阻板厚度的增大而提高,而随着梯度波阻板埋深的增大而降低.     

功能梯度材料明德林矩形微板的热弹性阻尼

功能梯度材料微板谐振器热弹性阻尼的建模和预测是此类新型谐振器热?弹耦合振动响应的新课题. 本文采用数学分析方法研究了四边简支功能梯度材料中厚度矩形微板的热弹性阻尼. 基于明德林中厚板理论和单向耦合热传导理论建立了材料性质沿着厚度连续变化的功能梯度微板热弹性自由振动控制微分方程. 在上下表面绝热边界条件下采用分层均匀化方法求解变系数热传导方程, 获得了用变形几何量表示的变温场的解析解. 从而将包含热弯曲内力的结构振动方程转化为只包含挠度振幅的偏微分方程. 然后,利用特征值问题在数学上的相似性,求得了四边简支条件下功能梯度材料明德林矩形微板的复频率解析解, 进而利用复频率法获得了反映谐振器热弹性阻尼水平的逆品质因子. 最后, 给出了材料性质沿板厚按幂函数变化的陶瓷?金属组分功能梯度矩形微板的热弹性阻尼数值结果. 定量地分析了横向剪切变形、材料梯度变化以及几何参数对热弹性阻尼的影响规律. 结果表明, 采用明德林板理论预测的热弹性阻尼值小于基尔霍夫板理论的预测结果, 而且两者的差别随着相对厚度的增大而变得显著.      

轴对称自由振动功能梯度材料微圆板中的热弹性阻尼

基于经典弹性薄板理论和单向耦合热传导理论,研究了材料性质沿厚度连续变化的功能梯度微圆板的热弹性阻尼特性．首先,考虑热力耦合效应,建立了功能梯度微圆板轴对称横向自由振动微分方程．然后,忽略温度梯度在面内的变化,建立了单向耦合变系数一维热传导方程．采用分层均匀化近似方法,将变系数热传导方程转化为一系列常系数的微分方程,利用上下表面的热边界条件和层间连续性条件获得了微圆板温度场解析解．将所得温度场代入微圆板的自由振动微分方程,得到了包含热弹性阻尼的复频率,从而获得了反映热弹性阻尼水平的逆品质因子．最后,针对材料性质沿板厚按幂函数变化的陶瓷-金属功能梯度微圆板,定量地分析材料梯度指数、几何尺寸、边界条件、温度环境等对微圆板热弹性阻尼的影响．     

饱和地基上含刚核弹性圆板的竖向振动分析

基于作者提出的饱和土弹性波动方程，研究了饱和地基上含刚核弹性圆板在竖向集中谐和力作用下的振动特性。首先应用Hankel积分变换求解该饱和土波动方程，然后按混合边值条件建立起一组描述含刚核弹性圆板振动的对偶积分方程，并将其化为第二类Fredholm积分方程进行数值求解。文末给出了含刚核弹性圆板在饱和地基上振动的阻抗函数随无量纲频率a0的变化曲线，并考察了土的渗透系数，弹性板含刚域的大小以及板的柔度等参数对阻抗函数的影响，得出了一些有意义的结论。     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

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 与 的计算表达式;进而,假设 向正应力在厚度方向上均匀分布,推导了弹性基础 向位移衰减函数 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,证明了本文理论的正确性;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 与体积分数指数 对薄板挠度分布的影响。     

A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation

This paper is motivated by the lack of studies in the technical literature concerning to the three-dimensional vibration analysis of bi-directional FG rectangular plates resting on two-parameter elastic foundations. The formulations are based on the three-dimensional elasticity theory. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. This paper presents a novel 2-D six-parameter power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. Various material profiles along the thickness and in the in-plane directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori-Tanaka scheme. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. Some new results for natural frequencies of the plates are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D FGM.     

Semi-analytical solution for three-dimensional vibration of thick continuous grading fiber reinforced (CGFR) annular plates on Pasternak elastic foundations with arbitrary boundary conditions on their circular edges

In this paper free vibration of continuous grading fiber reinforced (CGFR) annular plates on an elastic foundation, based on the three-dimensional theory of elasticity, for different boundary conditions at the circular edges is investigated. The foundation is described by the Pasternak or two-parameter model. The CGFR annular plates have an arbitrary variation of fiber volume fraction in the thickness direction. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. Besides, results for CGFR plate with arbitrary variation of fiber volume fraction in the thickness direction of the plate are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on an elastic foundation and comparison with similar discrete laminated composite plate. The interesting and new results show that non-dimensional natural frequency parameters of a functionally graded fiber volume fraction is larger than that of a discrete laminated and close to that of a 2-layer. The new results can be taken as the benchmark solutions for those from numerical methods and future researches.     

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.     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Nonlinear dynamic response of nanotube-reinforced composite plates resting on elastic foundations in thermal environments

This paper presents an investigation on the nonlinear dynamic response of carbon nanotube-reinforced composite (CNTRC) plates resting on elastic foundations in thermal environments. Two configurations, i.e., single-layer CNTRC plate and three-layer plate that is composed of a homogeneous core layer and two CNTRC surface sheets, are considered. The single-walled carbon nanotube (SWCNT) reinforcement is either uniformly distributed (UD) or functionally graded (FG) in the thickness direction. The material properties of FG-CNTRC plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion that includes plate-foundation interaction are solved by a two-step perturbation technique. Two cases of the in-plane boundary conditions are considered. Initial stresses caused by thermal loads or in-plane edge loads are introduced. The effects of material property gradient, the volume fraction distribution, the foundation stiffness, the temperature change, the initial stress, and the core-to-face sheet thickness ratio on the dynamic response of CNTRC plates are discussed in detail through a parametric study.     

Nonlinear dynamic analysis of moving bilayer plates resting on elastic foundations

The aim of this study is to investigate the dynamic response of axially moving two-layer laminated plates on the Winkler and Pasternak foundations. The upper and lower layers are formed from a bidirectional functionally graded(FG) layer and a graphene platelet(GPL) reinforced porous layer, respectively. Henceforth, the combined layers will be referred to as a two-dimensional(2D) FG/GPL plate. Two types of porosity and three graphene dispersion patterns, each of which is distributed through the plate thickness,are investigated. The mechanical properties of the closed-cell layers are used to define the variation of Poisson's ratio and the relationship between the porosity coefficients and the mass density. For the GPL reinforced layer, the effective Young's modulus is derived with the Halpin-Tsai micro-system model, and the rule of mixtures is used to calculate the effective mass density and Poisson's ratio. The material of the upper 2D-FG layer is graded in two directions, and its effective mechanical properties are also derived with the rule of mixtures. The dynamic governing equations are derived with a first-order shear deformation theory(FSDT) and the von Kármán nonlinear theory. A combination of the dynamic relaxation(DR) and Newmark's direct integration methods is used to solve the governing equations in both time and space. A parametric study is carried out to explore the effects of the porosity coefficients, porosity and GPL distributions, material gradients, damping ratios, boundary conditions, and elastic foundation stiffnesses on the plate response. It is shown that both the distributions of the porosity and graphene nanofillers significantly affect the dynamic behaviors of the plates. It is also shown that the reduction in the dynamic deflection of the bilayer composite plates is maximized when the porosity and GPL distributions are symmetric.