基于等几何有限元法的功能梯度微板热力耦合屈曲预测

基于修正偶应力理论和Kirchhoff板理论,建立了功能梯度微板热力耦合屈曲等几何有限元模型。该模型仅包含一个材料尺度参数,能够描述尺度效应现象,且满足修正偶应力理论的高阶连续性要求。基于虚功原理推导了功能梯度微板热力耦合屈曲等几何有限元方程。通过对板的典型算例分析,讨论了材料尺度参数、边长比及梯度指数对板稳定性的影响。结果表明,本文模型预测的屈曲载荷总是大于宏观理论的结果,即捕捉到了尺度效应现象;随着临界屈曲力的增加,临界屈曲热载荷线性减少;此外,边长比和梯度指数也对微板的稳定性产生一定影响。     

基于精化锯齿理论的功能梯度夹心微板静弯曲模型

基于精化锯齿理论和新修正偶应力理论,建立了能够准确预测功能梯度夹心微板挠度、位移和应力的静弯曲模型。为了描述微板不同方向上的尺度效应,将两个正交材料尺度参数引入本文模型。以受双向正弦载荷作用的简支板为例,探究了夹心微板弯曲行为中尺度效应对结构刚度的影响。算例结果表明,当微板几何参数与材料尺度参数接近时,基于本文模型所测微板的最大弯曲挠度、局部位移和应力均小于传统精化锯齿理论给出的结果,捕捉到了尺度效应;尺度效应随着微板几何尺寸的增大而逐渐减弱,当微板几何尺寸远大于材料尺度参数时,尺度效应消失。此外,板的跨厚比和功能梯度变化指数也会对尺度效应产生一定影响。     

碳纳米管增强型复合材料功能梯度板的自由振动模型与尺度效应

基于一种新的各向异性修正偶应力理论,建立了碳纳米管增强复合材料功能梯度板的自由振动模型。该模型能够描述尺度效应,且仅包含一个尺度参数。基于一阶剪切变形理论和哈密顿原理推演了板的运动微分方程,并以四边简支板为例给出了自振频率的解析解。讨论了板的几何尺寸、碳纳米管体分比含量和分布方式等因素对板的自振频率的影响。结果表明,本文模型所预测的板的自振基频总是高于经典弹性理论的Mindlin板模型的预测结果,两者间的差异在板的几何尺寸接近尺度参数的值时非常明显,且会随着板的几何尺寸的增大而逐渐消失。     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论,建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件,并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明,在材料几何尺寸较小时,本文模型预测到的屈曲载荷明显大于传统理论的结果,有效地反映了尺度效应。几何尺寸较大时,尺度效应消失,本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

尺度依赖的静电驱动各向异性微板的Pull-in效应

在新修正偶应力理论的基础上建立了一种可用于分析静电驱动各向异性微板的尺度依赖模型。模型中包含有两个材料尺度参数,在考虑材料宏观各向异性的同时也考虑了微观各向异性程度对结构Pull-in特性的影响。通过虚功原理推导了静电驱动微板的非线性控制方程并显式地给出了Pull-in电压和挠度的表达式。算例结果表明:本文模型所预测的Pull-in电压和挠度分别大于和小于经典宏观板理论的预测结果,即反映了微尺度结构下的尺度效应。尺度效应的影响在板厚度与材料尺度参数接近时逐渐增加,而随着两者比值的增加,该影响逐渐减弱,最终可以忽略不计。此外,本文也讨论了初始间隙d对Pull-in电压以及尺度效应的影响。结果表明随着d的增加,Pull-in电压随之增大,而Pull-in挠度变化不大。     

考虑尺度效应的微梁静态弯曲数值分析

基于修正偶应力理论及表面弹性理论,本文提出了一种新的双曲线剪切变形梁模型,用于均匀微尺度梁的静态弯曲分析。该理论可以直接利用本构关系获得横向剪切应力,满足梁顶部和底部的无应力边界条件,避免了引入剪切修正因子。根据广义Young-Laplace方程建立了梁的内部与表面层的应力连续性条件,单一的变量场可以描述梁的位移模式。通过在位移场中考虑表面层厚度以及表面层的应力连续条件,可以使新模型能够更准确地预测微尺寸和表面能相关的尺度效应。通过Hamilton原理推导出了梁的控制方程和边界条件。应变能除了考虑经典弹性理论,还要考虑微结构效应和表面能。Navier-type的解析解适用于简支边界条件,而基于拉格朗日插值的微分求积法(DQEM)可以研究在不同边界条件下的力学响应。把该数值解与Navier方法得出的解析解作了对比,得出:微尺度梁在考虑表面能或微尺寸效应、不同载荷和梁高变化下的响应一致;当不考虑微结构相关性和表面能效应时,该模型退化为经典的欧拉梁模型。     

正交各向异性功能梯度材料反平面裂纹尖端应力场

采用积分变换-对偶积分方程方法,研究了正交各向异性功能梯度材料反平面裂纹问题.文中假定材料沿两个主轴方向的剪切模量成比例按双参数梯度模型变化,通过求解对偶积分方程并考虑变形Bessel函数的渐近特性,推导出了裂纹尖端应力场.最后考察了材料非均匀性及正交性对应力强度因子的影响.     

考虑尺度效应的夹层楔形多孔梁动力学分析

基于修正偶应力理论, 研究了具有大范围旋转中心刚体-功能梯度夹层Euler-Bernoulli楔形多孔柔性微梁系统的动力学特性.楔形梁是中间层为不完全功能梯度层, 两表层为均质材料的功能梯度夹层结构, 它可以减小传统夹层结构由于层与层之间材料属性的不同导致脱粘类型损伤的影响.采用假设模态法描述变形, 考虑具有捕捉动力刚化效应的非线性耦合项, 计及von Kármán几何非线性应变, 运用第二类Lagrange方程, 导出了适用于较大变形的高次刚柔耦合动力学方程.对在平面内做大范围运动的中心刚体-功能梯度夹层Euler-Bernoulli楔形多孔微梁的动力学特性进行了详细研究.研究表明: 功能梯度夹层楔形梁表层结构高度、旋转角速度、功能梯度幂指数、尺度参数、孔隙度以及各层结构的体积分数对系统的动力学特性都有很大的影响; 功能梯度夹层楔形梁综合了功能梯度直梁和楔形梁的特性, 其相对于功能梯度直梁的固有频率增大, 同时使得孔隙度对结构固有频率变化趋势的影响不再与功能梯度直梁相同; 由于柔性梁变形能中具有横向与轴向的耦合势能, 系统在稳态下的平衡位置发生了迁移现象; 系统随着尺度参数的变化发生了频率转向与振型转换.      

弹性支撑功能梯度压电多孔微圆柱壳的自由振动分析

旨在研究热-力-电载荷下弹性支撑功能梯度压电多孔微圆柱壳的自由振动。首先,建立弹性支撑功能梯度压电多孔微圆柱壳动力学模型;然后,应用三阶剪切变形壳体理论和修正的偶应力理论,推出弹性支撑功能梯度压电多孔微圆柱壳模态频率的解析解;最后,通过数值算例分析了微圆柱壳模态频率的影响因素。结果表明：Pasternak弹性支撑比Winkler弹性支撑更有利于提高微圆柱壳的模态频率;改变弹性支撑的刚度系数、轴向力、外加电压、孔隙分布、材料体积分数指数和结构尺寸可调节微圆柱壳的模态频率;孔隙体积分数越大,温度或轴向力对模态频率的影响越大,而电压对模态频率的影响则越小;不同材料指数下,增大孔隙体积分数对模态频率的影响趋势不同;弹性支撑会减弱温度、轴向力和电压对模态频率的影响,对薄圆柱壳或短圆柱壳模态频率的影响较为显著。     

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.     

Instability of functionally graded micro-beams via micro-structure-dependent beam theory

This paper focuses on the buckling behaviors of a micro-scaled bi-directional functionally graded (FG) beam with a rectangular cross-section, which is now widely used in fabricating components of micro-nano-electro-mechanical systems (MEMS/NEMS) with a wide range of aspect ratios. Based on the modified couple stress theory and the principle of minimum potential energy, the governing equations and boundary conditions for a micro-structure-dependent beam theory are derived. The present beam theory incorporates different kinds of higher-order shear assumptions as well as the two familiar beam theories, namely, the Euler-Bernoulli and Timoshenko beam theories. A numerical solution procedure, based on a generalized differential quadrature method (GDQM), is used to calculate the results of the bi-directional FG beams. The effects of the two exponential FG indexes, the higher-order shear deformations, the length scale parameter, the geometric dimensions, and the different boundary conditions on the critical buckling loads are studied in detail, by assuming that Young’s modulus obeys an exponential distribution function in both length and thickness directions. To reach the desired critical buckling load, the appropriate exponential FG indexes and geometric shape of micro-beams can be designed according to the proposed theory.     

Hamilton体系下功能梯度梁的热冲击动力屈曲分析

在Hamilton体系下,基于Euler梁理论研究了功能梯度材料梁受热冲击载荷作用时的动力屈曲问题;将非均匀功能梯度复合材料的物性参数假设为厚度坐标的幂函数形式,采用Laplace变换法和幂级数法解析求得热冲击下功能梯度梁内的动态温度场:首先将功能梯度梁的屈曲问题归结为辛空间中系统的零本征值问题,梁的屈曲载荷与屈曲模态分别对应于Hamilton体系下的辛本征值和本征解问题,由分叉条件求得屈曲模态和屈曲热轴力,根据屈曲热轴力求解临界屈曲升温载荷。给出了热冲击载荷作用下一类非均匀梯度材料梁屈曲特性的辛方法研究过程,讨论了材料的梯度特性、结构几何参数和热冲击载荷参数对临界温度的影响。     

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of the major threats to their service safety in engineering applications. This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams. First, based on the Euler-Bernoulli beam theory and von Kármán geometric nonlinearity, a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang's two-variable method is formulated. Second, an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finally, based on the free energy theorem, it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermore, the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.     

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higher...     

新修正偶应力理论Reddy型层合板稳定分析

基于新修正偶应力理论建立了一个Reddy型复合材料层合板稳定性模型。该理论中曲率张量不对称,而偶应力矩张量对称。Reddy型层合板模型能够满足横向剪切应力为0的自由表面条件,而且横向剪切为二次函数,避免了常剪力一阶理论需要引入的剪力修正系数。为了便于工程应用,通过虚功原理推导了只含纤维材料尺度参数正交铺设的Reddy型层合板偶应力模型的稳定性方程,并以微尺度正交铺设四边简支层合方板为例,分析了不同铺设角和轴向载荷作用时临界载荷的细观尺度效应,并且与一阶剪切变形和Kirchhoff板理论结果对比。结果表明,本文建立的新修正偶应力Reddy型层合板模型更适合分析较厚的复合材料层合板稳定性的尺度效应。     

Nonlocal thermal buckling and postbuckling of functionally graded graphene nanoplatelet reinforced piezoelectric micro-plate

This paper analyzes the nonlocal thermal buckling and postbuckling behaviors of a multi-layered graphene nanoplatelet(GPL) reinforced piezoelectric micro-plate. The GPLs are supposed to disperse as a gradient pattern in the composite micro-plate along its thickness. The effective material properties are calculated by the Halpin-Tsai parallel model and mixture rule for the functionally graded GPL reinforced piezoelectric(FG-GRP) micro-plate. Governing equations for the nonlocal thermal buckling a...     

Two-dimensional magneto-thermoelastic interactions in a micropolar functionally graded solid

Abstract

The present work is concerned with the 2D deformation in a nonhomogeneous, isotropic, micropolar, magneto-thermoelastic medium in the context of Lord-Shulman theory as a result of an inclined load. The inclined load is supposed to be a linear combination of normal load and tangential load. Material properties are assumed to be graded in x-direction. Normal mode technique is proposed to obtain the analytical expressions for the temperature field, displacement components, and stresses. These are also calculated numerically and depicted graphically to observe the variations of the considered physical variables.

Communicated by Seonho Cho.