功能梯度与均匀圆板静动态解之间的相似转换关系

基于经典板理论,研究了功能梯度材料圆板的轴对称弯曲、屈曲和自由振动解与相应的均匀材料圆板解之间的转换关系.通过消去拉-弯耦合项得到了以挠度函数表示的功能梯度圆板的弯曲、屈曲和自由振动控制方程.分析功能梯度圆板与均匀圆板的控制方程之间的相似性,得到了功能梯度材料圆板与均匀圆板的解之间解的相似转换关系,在假定FGM圆板的材料性质沿厚分别以幂函数和指数函数的度变规律后,给出了相应的转换系数的解析表达式.该系数集中反映了功能梯度圆板的材料非均匀性.在已知均匀材料圆板轴对称解的条件下,可将功能梯度材料圆板轴对称问题的求解转化为相似转换系数的计算问题.这一方法可为非均匀板的求解提供了十分便捷有效的途径,而且便于工程应用.     

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。     

剪切变形对FGM圆板轴对称屈曲的影响

基于一阶剪切变形板理论，推导了功能梯度材料圆形板在边界面内均布压力作用下的轴对称屈曲方程。在推导过程中，忽略了前屈曲耦合变形。利用一阶板理论与经典板理论屈曲方程之间在数学形式上的相似性，得到了一阶板理论下功能梯度材料圆板与经典板理论下均匀圆板临界屈曲载荷之间的解析关系。利用这个解析关系，可以直接从已有的较为简单的经典理论的结果，获得一阶板理论下功能梯度材料板的临界屈曲载荷。     

功能梯度材料矩形中厚板的受压/热致屈曲

考虑材料组份沿板厚度方向按幂律变化的情形,研究了温度均匀变化时固支功能梯度材料(FGM)矩形中厚板的受压屈曲、热致屈曲和考虑热／机械预应力时的屈曲问题,给出了基于Reddy高阶剪切理论研究板屈曲荷载和屈曲临界温度的半解析数值方法．并以Si3N4／SUS304板为例考虑了材料组份、预加应力、横向剪切变形及面内位移约束条件等对FGM板屈曲承载能力的影响．     

基于均匀化转换方法的FGM圆柱壳静水压力下临界压力预测研究

依据经典Flügge壳体理论，利用功能梯度材料(FGM)和均匀材料物理性质和力学行为相似性的均匀化转换计算方法，研究了静水压力下FGM圆柱壳临界压力的预测方法。针对水下FGM圆柱壳耦合系统的振动问题，考虑流体影响，采用波动法推导出相应的振动方程，使用了牛顿迭代法，以确定在静水压力下FGM圆柱壳的固有频率。根据临界载荷与固有频率为零的载荷水平线性相关性，运用拟合曲线法和均匀化转换后的公式法对静水压力下FGM圆柱壳临界压力进行了预测分析，并讨论了FGM圆柱壳各项参数对静水压力下FGM圆柱壳临界压力的影响。结果表明，FGM圆柱壳的材料弹性模量E合值、几何尺寸h/R和L/R,以及不同边界条件改变对临界压力影响较大。通过对多组算例的对比分析，证明了本研究方法的正确性和有效性。使用该方法进行预测的精度高，计算量小，能够为非均匀结构力学行为的分析提供新的研究途径。     

受面内双向荷载蜂窝材料的弹性屈曲模式及其影响因素

基于压弯耦合梁柱稳定理论,分析了蜂窝材料在面内双轴应力作用下的弹性屈曲行为,得到了蜂窝材料在不同应力荷载作用下发生两种屈曲模式转换的临界条件和对应的屈曲临界荷载.讨论了孔穴几何参数对此转换条件和临界荷载的影响,并用ANSYS软件进行了有限元模拟.结果表明理论分析与数值模拟吻合很好.     

非保守力下温度场中FGM圆板的非线性力学行为

研究了温度场中非保守功能梯度材料(FGM)圆板的非线性力学行为。基于经典板理论,推导了受非保守力作用的FGM圆板在温度场中的控制微分方程。采用打靶法分析了由陶瓷二氧化锆和金属钛合金两相材料组成的非保守FGM圆板在均匀和非均匀升温场中的非线性力学行为。给出了不同均匀升温和非均匀升温场下,FGM圆板在非保守载荷作用下的平衡路径和平衡构形。分析并讨论了均匀和非均匀升温、材料梯度指数对非保守圆板过屈曲和弯曲行为的影响。结果表明:温度场中,非保守FGM圆板发生弯曲而纯陶瓷圆板会发生过屈曲行为;当梯度指数p=2,非保守载荷q=52时,均匀升温场中非保守圆板的变形大于非均匀升温场中非保守圆板的变形。     

轻质金属点阵夹层板热屈曲临界温度分析

本文针对均匀温度场下四边简支和四边固支金属点阵夹层板的临界热屈曲温度进行了求解和参数影响分析。将点阵夹芯等效为均匀连续体,并且将夹层板的剪切刚度近似为点阵夹芯的抗剪切刚度,忽略夹芯的抗弯刚度且认为夹层板主要由面板来提供抗弯刚度。对于无法获得解析解的四边固支条件,通过对未知变量进行双傅里叶展开的方法求解了Ressiner夹层板模型的临界屈曲温度,理论分析结果与有限元计算结果吻合良好。进一步分析了不同边界条件、点阵胞元构型、点阵材料相对密度、面板厚度等对临界屈曲温度的影响规律。     

中厚圆柱壳在局部荷载下有限变形与屈曲分析

在柱壳的有限元计算中，采用Mindlin八结点杂交壳单元和增量荷载法，基于选择积分，缩减积分及完全积分三种积分模式编制了分层计算各种厚度板壳的有限元程序FEAM，并对中厚圆柱壳在局部法向均布荷载作用下的弹塑性有限变形和屈曲问题进行了分析和计算，算例表明，利用FEAM可对壳体屈曲的临界荷载及屈曲后结构的承载与形状改变作定性与定量的分析.     

反对称横向荷载作用下钢梁的弹性弯扭屈曲研究

为了建立反对称横向荷载作用下双轴对称截面简支钢梁弹性弯扭屈曲的设计理论,考虑荷载比例系数ψ的影响,推导了反对称横向荷载作用下钢梁的弯扭屈曲总势能方程。采用Rayleigh-Ritz法得到了反对称横向荷载作用下钢梁弹性临界弯矩M_(cr)的通用计算式以及系数C_1、C_2的计算式,并总结了荷载比例系数ψ对临界弯矩的影响规律。采用有限元法对本文理论公式进行了验证,当0≤ψ≤4时,临界弯矩M_(cr)的理论解与有限元解吻合良好;当ψ4时,临界弯矩收敛于ψ=4时的值。在此基础上,通过线性回归分析,拟合出了等效弯矩系数C_b与荷载比例系数ψ以及跨长影响系数ξ之间的近似关系式,C_b的近似解与有限元解吻合较好,最大误差为6.5%,大部分工况下误差控制在5%以内。该拟合公式适用性较强,精度较高。     

CLASSICAL AND HOMOGENIZED EXPRESSIONS FOR BUCKLING SOLUTIONS OF FUNCTIONALLY GRADED MATERIAL LEVINSON BEAMS

The relationship between the critical buckling loads of functionally graded material(FGM) Levinson beams(LBs) and those of the corresponding homogeneous Euler-Bernoulli beams(HEBBs) is investigated. Properties of the beam are assumed to vary continuously in the depth direction. The governing equations of the FGM beam are derived based on the Levinson beam theory, in which a quadratic variation of the transverse shear strain through the depth is included.By eliminating the axial displacement as well as the rotational angle in the governing equations,an ordinary differential equation in terms of the deflection of the FGM LBs is derived, the form of which is the same as that of HEBBs except for the definition of the load parameter. By solving the eigenvalue problem of ordinary differential equations under different boundary conditions clamped(C), simply-supported(S), roller(R) and free(F) edges combined, a uniform analytical formulation of buckling loads of FGM LBs with S-S, C-C, C-F, C-R and S-R edges is presented for those of HEBBs with the same boundary conditions. For the C-S beam the above-mentioned equation does not hold. Instead, a transcendental equation is derived to find the critical buckling load for the FGM LB which is similar to that for HEBB with the same ends. The significance of this work lies in that the solution of the critical buckling load of a FGM LB can be reduced to that of the HEBB and calculation of three constants whose values only depend upon the throughthe-depth gradient of the material properties and the geometry of the beam. So, a homogeneous and classical expression for the buckling solution of FGM LBs is accomplished.     

Size-dependent shear buckling response of FGM skew nanoplates modeled via di?erent homogenization schemes

The size e?ects on the shear buckling behaviors of skew nanoplates made of functionally graded materials(FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated e?ective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the signi?cance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less signi?cant for a higher value of the material property gradient index. Furthermore,among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the di?erence between them reduces by increasing the aspect ratio of the skew nanoplate.     

Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements

Three-dimensional enriched finite elements are used to compute mixed-mode stress intensity factors (SIFs) for three-dimensional cracks in elastic functionally graded materials (FGMs) that are subject to general mixed-mode loading and constraint conditions. The method, which advantageously does not require special mesh configuration/modifications and post-processing of finite element results, is an enhancement of previous developments applied so far on isotropic homogeneous and isotropic interface cracks. The spatial variation of FGM material properties is taken into account at the level of element integration points. To validate the developed method, two- and three-dimensional mixed-mode fracture problems are selected from the literature for comparison. Two-dimensional cases include: inclined central crack in a large FGM medium under uniform tensile strain and stress loadings, a slanted crack in a finite-size FGM plate under exponentially varying tensile stress loading and an edge crack in a finite-size plate under shear traction load. The three-dimensional example models a deflected surface crack in a finite-size FGM plate under uniform tensile stress loading. Comparisons between current results and those from analytical and other numerical methods yield good agreement. Thus, it is concluded that the developed three-dimensional enriched finite elements are capable of accurately computing mixed-mode fracture parameters for cracks in FGMs.     

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogenization schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.     

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.     

Dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells,under combined axial compression and external pressure

FGM components are constructed to sustain high temperature gradients. There are many applications where the FGM components are vulnerable to transient thermal shocks. If a component is already under compressive external loads (e.g. under a combination of axial compression and external pressure), the mentioned thermal shocks will cause the component to exhibit dynamic behavior and in some cases may lead to buckling. On the other hand, a preheated FGM component may undergo dynamic mechanical loads. Only static thermal buckling investigations were developed so far for the FGM shells. In the present paper, dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment, with temperature-dependent properties are presented. The general form of Green’s strain tensor in curvilinear coordinates and a high order shell theory proposed already by the author are used. Instead of using semi-analytical solutions that rely on the validity of the separation of variables concept, the complicated nonlinear governing equations are solved using the finite element method. Buckling load is detected by a modified Budiansky criterion proposed by the author. The effects of temperature-dependency of the material properties, volume fraction index, load combination, and initial geometric imperfections on the thermo-mechanical post-buckling behavior of a shell with two constituent materials are evaluated. The results reveal that the volume fraction index and especially, the differences between the thermal stresses created in the outer and the inner surfaces may change the buckling behavior. Furthermore, temperature gradient and initial imperfections have less effect on buckling of a shell subjected to a pure external pressure.     

Buckling of a coating strip of finite width bonded to elastic half-space

A solution for buckling of a stiff strip of finite width bonded to a compliant elastic half-space and subjected to uniform axial compression is presented. Approximate semi-analytical and finite element solutions are obtained and compared with a two-dimensional case of a plate on elastic foundation. The comparison demonstrates that the two-dimensional solution can be applied to predict the buckling wavelength and critical compressive strain when the width of the strip is equal to or larger than the buckling wavelength. For narrow strips, the wavelength is smaller and critical strain is higher than that of a plate on foundation.     

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。