碳纳米管增强型功能梯度板的屈曲预测

基于一种新的修正偶应力理论,建立了碳纳米管（CNTs）增强型功能梯度板（CNTs/FGP）的屈曲模型。基于最小势能原理和一阶剪切变形理论,推导了该种板模型的平衡微分方程和相应的边界条件,并以四边简支方板的屈曲问题为例,讨论了材料尺度参数、CNTs的体积分数及4种不同CNTs分布形式对CNTs/FGP临界屈曲载荷的影响。结果表明：采用本文模型预测的CNTs/FGP的临界屈曲载荷总是大于传统宏观理论的预测结果,两种理论结果间的差距随着板几何尺寸的减小而逐渐增大;CNTs体积分数的少量增加,即可使板的临界屈曲载荷有明显的提升;CNTs的不同分布形式对临界屈曲载荷有显著的影响,在工程设计中应予以关注。     

基于一种新修正偶应力理论的平面正交各向异性功能梯度梁静弯曲模型及尺度效应

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁模型。模型中包含两个材料尺度参数,因此能够分别描述在两个正交方向上由尺度效应带来的不同大小弯曲刚度增强。基于最小势能原理推导了平衡方程和边界条件,并以自由端受集中载荷作用的悬臂梁为例给出了弯曲问题的解析解。该梁模型的控制方程以及解的形式和经典梁模型是一致的,只是在刚度项中增加了一项和尺度效应有关的项。算例结果表明:采用本文模型所预测的梁挠度总是小于经典理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的减小而增大,并在梁的几何尺寸远大于尺度参数时逐渐消失。     

考虑尺度依赖的平面正交各向异性功能梯度微梁的自由振动分析

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁的自由振动模型。模型中包含两个材料尺度参数,能够分别描述两个正交方向上不同程度的尺度效应。当梁的几何尺寸远大于材料尺度参数时,本文模型亦可自动退化为相应的传统宏观模型。基于哈密顿原理推导了运动控制方程并以简支梁的自由振动为例分析了几何尺寸、功能梯度变化指数等对尺度效应产生的影响。算例结果表明:采用本文模型所预测的梁自振频率总是大于传统理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的增大而逐渐减弱并在几何尺寸远大于尺度参数时消失;高阶自振频率所体现出的尺度效应较低阶自振频率更加明显。此外,功能梯度变化指数对尺度效应也有一定的影响。     

基于正弦剪切变形理论的功能梯度材料三明治微梁的静动态特性

基于修正的偶应力理论和正弦剪切变形梁理论,研究了功能梯度材料三明治微梁的静态弯曲和自由振动行为。考虑两种不同类型的功能梯度材料三明治微梁,根据哈密顿变分原理建立其静动态力学行为的控制方程,应用Navier解法,得到了简支边界条件下弯曲变形和振动频率的解析解,同时,给出了固支等边界条件时的里兹法求解过程。数值算例表明,功能梯度三明治微梁的静动态力学行为具有明显的尺度效应,微梁的无量纲厚度、功能梯度指数、长厚比和结构形式等因素对其静动态响应有很大影响,相关结果和规律对功能梯度材料三明治微梁的结构设计和性能优化等实际工程应用具有一定的指导意义。      

Modeling of functionally graded smart plates with gradient elasticity effects

ABSTRACT

In this article, the equations of motion for functionally graded plates with surface-mounted piezoelectric layers, while accounting for the gradient elasticity through the modified couple stress model and linear piezoelectricity, are derived using Hamilton’s principle. The formulation includes the coupling between mechanical deformations and the charge equations of electrostatics. The mathematical model developed herein is an equivalent single layer theory for mechanical displacement field and the potential functions. The in-plane displacements are assumed to vary as cubic functions of the thickness coordinate while the transverse displacement is assumed to vary as a quadratic function of the thickness coordinate through plate thickness. The potential function is assumed as the combination of half cosine variation of electric potential and linear variation of applied voltage on outer surfaces. The approach described here is that standard plate models can be enhanced to include the coupling between the charge equations and the mechanical deformations as well as the size dependent effect of micro- and nano-scale structures. An analytical solution of the developed model is presented using the Navier solution technique. A parametric study is performed to study the effect of material variation through thickness of plates, length scale parameters to capture the size-dependent effects, and thickness ratio between piezoelectric layers and the whole plate.     

Bending-stretching analysis of thick functionally graded micro-plates using higher-order shear and normal deformable plate theory

In the present article, higher-order shear and normal deformable plate theory together with modified couple stress theory are developed to study the bending analysis of thick functionally graded rectangular micro-plates. One material length scale parameter is used for capturing the size effects. Utilizing the variational approach and also a principle of virtual displacement, a new form of equilibrium equations and the corresponding boundary conditions are derived. It is assumed that material properties vary through the thickness according to the power law function. Finally, an analytical solution for the bending problem of a simply supported FG rectangular micro-plate is presented.     

TiC-Ni梯度功能材料的优化设计

对ＴｉＣ－Ｎｉ梯度功能材料在制备过程中的残余热应力进行了计算机有限元模拟,考察了梯度组成分布指数对热应力大小,最大热应力发生的位置以及纯陶瓷ＴｉＣ侧热应力状态的影响,综合分析了热应力的大小和分布,得到了缓和制备热应力的梯度组成分布指数Ｐ＝１．０的优化设计结果。     

The modified couple stress functionally graded cylindrical thin shell formulation

In this article, the functionally graded (FG) cylindrical thin shell formulation is developed by using modified couple stress theory. The equations of motion and classical and nonclassical boundary conditions are extracted based on Hamilton's principle. As a special case, the equations of motion in conjunction with the boundary conditions for simply supported FG cylindrical shell are obtained, and then Navier solution procedure is used for analysis free vibration of nano shell. Afterwards, the influences of different parameters like length scale parameter, distribution of FG properties, and length to radius ratio on dimensionless natural frequency are investigated and compared with classical theory.     

A nonlinear modified couple stress-based third-order theory of functionally graded plates

In this paper a general nonlinear third-order plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally graded material plates) is presented using the principle of virtual displacements. A detailed derivation of the equations of motion, using Hamilton’s principle, is presented, and it is based on a modified couple stress theory, power-law variation of the material through the thickness, and the von Kármán nonlinear strains. The modified couple stress theory includes a material length scale parameter that can capture the size effect in a functionally graded material. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity, microstructure dependent size effect, and material gradation through the thickness are specialized to classical and shear deformation plate theories available in the literature. The theory presented herein also can be used to develop finite element models and determine the effect of the geometric nonlinearity, microstructure-dependent size effects, and material grading through the thickness on bending and postbuckling response of elastic plates.     

Transient thermal stress analysis of an edge crack in a functionally graded material

An edge crack in a strip of a functionally graded material (FGM) is studied under transient thermal loading conditions. The FGM is assumed having constant Young's modulus and Poisson's ratio, but the thermal properties of the material vary along the thickness direction of the strip. Thus the material is elastically homogeneous but thermally nonhomogeneous. This kind of FGMs include some ceramic/ceramic FGMs such as TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs such as zirconia/nickel and zirconia/steel. A multi-layered material model is used to solve the temperature field. By using the Laplace transform and an asymptotic analysis, an analytical first order temperature solution for short times is obtained. Thermal stress intensity factors (TSIFs) are calculated for a TiC/SiC FGM with various volume fraction profiles of the constituent materials. It is found that the TSIF could be reduced if the thermally shocked cracked edge of the FGM strip is pure TiC, whereas the TSIF is increased if the thermally shocked edge is pure SiC.     

Transient waves in plates of functionally graded materials

A numerical method is proposed for analysing transient waves in plates of functionally graded material (FGM) excited by impact loads. The material properties of the FGM plate have a gradient in the thickness direction and are anisotropic in the plane of the plate. In the present method, the FGM plate is divided into layer elements in the thickness direction. For an accurate modelling of the variation of the material property of FGM plates, it is expressed by second‐order polynomials in the thickness direction within an element. This can further reduce the number of elements to obtain more accurate results effectively. The principle of virtual work is used to develop approximate dynamic equilibrium equations. The displacement response is determined by employing the Fourier transformation and the modal analysis. As examples, the displacement response of FGM plates excited by line, point and distributed loads is calculated. The computations have shown the efficiency of the present method. Copyright © 2001 John Wiley & Sons, Ltd.     

功能梯度铁电铁磁复合材料弯曲模态下的磁电效应

基于磁致伸缩相与压电相的本构方程,应用弹性力学的方法,建立了功能梯度铁电铁磁复合材料弯曲模态下的磁电耦合静态力学模型。假设铁电和铁磁材料的物理参数均为沿厚度方向的线性或指数函数,分析计算了由PZT作为铁电材料和CoFe2O4作为铁磁材料的双层复合材料的磁电效应。结果表明,在弯曲模态下,磁电电压系数出现两个峰值。负梯度的铁电(或铁磁)材料提高磁电效应,正梯度的铁电(或铁磁)材料降低磁电效应。同号梯度的铁电铁磁材料对磁电效应的影响更大。     

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

Small scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.     

多孔功能梯度梁的热-力耦合屈曲行为

采用经典欧拉梁理论和高阶三角剪切变形理论,研究了多孔功能梯度梁的热-力耦合屈曲行为。分析中考虑了材料物性与温度的相关性,采用含孔隙率的Voight混合模型描述了多孔功能梯度的材料属性。利用迭代算法求解结构在均匀、线性和非线性温升(考虑热传导效应)下的热-力耦合屈曲临界温度,讨论了材料非均匀参数、孔隙率和长细比等参数对屈曲临界温度的影响。ABAQUS数值模拟结果和文献对比结果验证了理论的可靠性,同时表明高阶剪切变形理论较经典欧拉梁理论精确。结果表明,功能梯度材料梁的热屈曲分析必须考虑物性与温度的相关性,否则可能高估热屈曲临界温度10%~30%;随着孔隙率增大,材料的等效弹性模量减少,即结构刚度有所弱化,但屈曲临界温度反而大大增高。     

Buckling of slender columns with functionally graded microstructures

The buckling of slender columns with functionally graded microstructures is studied. In such columns, the flexural modulus is varied in a controlled manner along the column length. The objective is to identify microstructures that maximize (and minimize) the critical buckling load when compared to a reference homogeneous column. Several microstructures are examined and a constraint is imposed so that the volume averaged flexural modulus remains the same in all columns. The buckling load is determined using both the linear perturbation analysis as well as the Rayleigh–Ritz method. A relationship between the material distribution and the corresponding mode shape is established.     

基于代理模型的功能梯度梁的材料特性参数反求

针对功能梯度材料参数的反求问题,提出了一种基于代理模型的反求方法。应用有限元软件建立功能梯度梁的波动响应模型,获得模型在激振力作用下的动态位移响应。通过试验设计选取合理的样本点,建立响应面模型代替有限元软件作为程序的正问题求解器,遗传算法作为反问题求解器,用添加不同噪声水平的峰值位移来模拟反求模型的实际输入作为整个算法的输入数据,最终获得材料的体积参数。以实际SiC-C功能梯度梁为算例,分析了单次正弦激振力下,有限元软件获得的动态响应,并根据获得的位移响应应用遗传算法来反求SiC-C梁的体积分数。算例验证了本方法的有效性。基于代理模型的反求方法避免了多次调用正问题求解器,提高了计算效率。      

Characteristics of waves in a functionally graded cylinder

An analytical‐numerical method is presented for analysing characteristics of waves in a cylinder composed of functionally graded material (FGM). In this method, the FGM cylinder is divided into a number of annular elements with three‐nodal‐lines in the wall thickness. The elemental material properties are assumed to vary linearly in the thickness direction so as to better model the spatial variation of material properties of FGM. The Hamilton principle is used to develop the dispersion equations for the cylinder, and the frequency and the group velocity are established in terms of the Rayleigh quotient. The method is applied to analyse several FGM cylinders, and its efficiency is demonstrated. Numerical results demonstrate that the ratio of radius to thickness has a stronger influence on the frequency spectra in the circumferential wave than on that in the axial wave, that negative group velocity presents at a range of smaller wave numbers and that the range varies as the wave normal and the ratio of radius to thickness of FGM cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

圆板状SiC/C功能梯度材料残余热应力特征有限元分析

功能梯度材料残余热应力的大小及分布对其性能有效发挥及长期稳定使用有着较大的负面影响,为了尽可能充分发挥材料性能,增加材料的使用寿命,需尽可能减小残余应力以及使其合理分布.本文采用ANSYS有限元分析软件对不同叠层工艺参数的等离子体第一壁候选材料--SiC/C功能梯度材料(FGM)的残余热应力进行了数值模拟,获得了使热应力有效缓和的较适宜的工艺参数,对实际研发制备目标材料也可提供一些理论参照.相关结果表明,适量增加梯度叠层数及中间梯度层厚度可逐步有效缓和残余热应力,同时,针对本文今后应用的仍以炭材料为主体的炭基陶瓷保护层复合SiC/C FGM而言,纯SiC层厚度应取较小值,而叠层成分分布指数应取0.8～1.0为宜.     

Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores

The bending response of functionally graded material (FGM) sandwich plates subjected to thermomechanical loads is investigated using a four-variable refined plate theory. A new type of FGM sandwich plate, namely, both FGM face sheets and an FGM hard core, is considered. Containing only four unknown functions, the governing equations are deduced based on the principle of virtual work and then these equations are solved via the Navier approach. Analytical solutions are obtained to predict the deflections and stresses of simply supported FGM sandwich plates. Benchmark comparisons of the solutions obtained for a degradation model (functionally graded face sheets and homogeneous cores) with ones computed by several other theories are conducted to verify the accuracy and efficiency of the present approach. The influences of volume fraction distribution, geometrical parameters, and thermal load on dimensionless deflections and normal and shear stresses of the FGM sandwich plates are studied.     

Development of cementless metal-backed acetabular cup prosthesis using functionally graded material

Metal backing has become widely used in acetaular cup design. A stiff backing for a polyethylene liner was initially believed to be mechanically favorable. Yet, recent studies of the load transfer around acetabular cups have shown that a stiff backing in fact generates higher stress peaks around the acetabular rim than full polyethylene cups, while reduces the stresses transferred at the central part of acetabulum causing stress shielding at the dome of acetabulum. To overcome these two problems, the aim of this study is to improve the design of cementless metal-backed acetabular cup using the two-dimensional functionally graded material concept through a finite element analysis and the optimization techniques. It is found that the optimal 2-D FGM model has three bioactive materials of hydroxyapatite, Bioglass and collagen. This optimal material reduces the stress shielding at the dome of acetabulum by 40% and 37% compared with stainless steel and titanium metal backing shell, respectively. However, using 2-D FGM model reduces the maximum interface shear stress in bone by 31% compared to titanium metal backing shell.