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.     

Low velocity impact analysis of sandwich plates with functionally graded face sheets

This article deals with the study of low velocity impact response on sandwich plates with functionally graded face sheets. High-order sandwich plate theory is improved by considering the in-plane stresses of the core that usually are ignored in the analysis of sandwich structures. A new approach is used to reduce the equations of motion from 27 equations to 15 equations and then solving them for both unsymmetric and symmetric sandwich plates. The model is also checked by finite element simulation and by comparing with other references for validation. A parametric study is done for various geometrical and mechanical properties.     

分布载荷作用下简支功能梯度夹层板的弯曲分析

研究了四边简支具有功能梯度芯材的夹层板在分布载荷作用下的弯曲问题。基于Reissner假设, 根据功能梯度材料的本构方程得出了应力、位移及内力的表达式, 得到功能梯度夹层板的平衡方程; 针对四边简支的边界条件, 通过将挠度 w 与横向剪力 Q_x、Q_y 用双三角级数展开的方法, 求解平衡方程。采用本文方法分别求解了均布载荷作用下、芯材弹性模量线性变化的功能梯度夹层板与芯材为均质各向同性材料的夹层板的弯曲挠度, 并通过与经典解及有限元解进行比较, 证明了本文方法的正确性。     

复杂边界条件功能梯度板三维分析的半解析方法

推导出适应功能梯度材料构件分析的半解析方法基本算式,并针对功能梯度构件的材料参数随空间坐标变化的特点,将材料参数纳入到力学方程中进行整体积分计算,从而编制统一程序计算不同边界条件下的板件问题.该法适应性强而又简洁高效,且不同于一般的半解析法,可采用一维离散,给出三维分析结果,是一种解决功能梯度构件力学性能分析的有效数值方法.文中用半解析法分析几种具有不同复杂边界条件的功能梯度板,给出了板件的力学量三维分布形态.     

由材料组分直接计算功能梯度复合材料开孔板宏观响应的跨尺度分析

建议一种新颖的功能梯度构件分析的细观元法, 给出了方法模型、基本算式及特点与功能。细观元法对构件的常规有限单元内部设置密集细观单元以反映材料组分梯度变化, 又通过协调条件将各细观元结点自由度转换为同一常规有限元自由度, 再上机计算。此法可实现材料细观结构到构件宏观响应的直接过渡分析, 而计算单元与自由度又等同于常规有限元, 为解决功能梯度构件宏观、细观跨尺度分析提供了一种有效工具。本文中直接从制备时给定材料组分分布出发计算构件宏观响应, 给出了不同开孔形状与数量功能梯度板的力学量三维分布形态。      

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.     

Three-dimensional solution of axisymmetric bending of functionally graded circular plates

Based on three-dimensional theory, this paper investigates the axisymmetric bending of transversely isotropic and functionally graded circular plates subject to arbitrarily transverse loads using the direct displacement method. The material properties can arbitrarily vary along the thickness of the plate. The transverse load is expanded in the Fourier–Bessel series and superposition principle is then used to obtain the total response based on the results of each item of the series. For one item of the series of the load, we assume the distributions of the displacements in the radial direction and therefore only the distributions of the displacements in thickness direction are required to find. If the material properties vary in an exponential law, the exact solutions can be obtained for elastic simple support and rigid slipping support, which are satisfied on the every point of the boundaries. Moreover, the analytical solutions are also presented for simply supported and clamped conditions, which are satisfied using Saint Venant principle. Simultaneously, through the layerwise method a semi-analytical solution is proposed for the case of arbitrary variation of the material properties. Finally the numerical examples are presented to verify the proposed method.     

Three-dimensional vibration analyses of functionally graded material rectangular plates with through internal cracks

The free vibrations of rectangular FGM plates with through internal cracks are investigated using the Ritz method. Three-dimensional elasticity theory is employed, and new sets of admissible functions for the displacement fields are proposed to enhance the effectiveness of the Ritz method in modeling the behaviors of cracked plates. The proposed admissible functions accurately describe the stress singularities at the fronts of the crack and display displacement discontinuities across the crack. The correctness and validity of the present approach are established through comprehensive convergence studies and comparisons with published results for homogeneous cracked plates, based on various plate theories. The locally effective material properties of FGM in the thickness direction are estimated by a simple power law. The effects of the volume fraction of the constituents of FGM and the thickness-to-length ratio on the frequencies are investigated. Frequency data for FGM square plates with three types of boundary conditions along the four side faces and with internal cracks of various crack lengths, positions and orientations are tabulated for the first time.     

Modeling of functionally graded sandwich shells accounting for variation in transverse displacement

In this study, a simple C₀ isoparametric finite element formulation based on higher-order shear deformation theory is presented for static analysis of functionally graded material sandwich shells (FGMSS). To characterize the membrane-flexure behavior observed in a functionally graded shell, a displacement field involving higher-order terms in in-plane and transverse fields is considered. The proposed kinematics field incorporates for transverse normal deformation, transverse shear deformation, and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the shell in an accurate sense. To develop the efficient C₀ formulation, the derivatives of transverse displacement are treated as independent field variables (nodal unknowns). Voigt's rule of mixture is employed to ascertain the mechanical properties of each layer's constituents along the thickness direction. A wide range of numerical problems are solved assuming various parameters: side-thickness ratio, curvature-side ratio, and gradation parameter, and their interactions with regard to static analysis of FGMSS are discussed in brief. Deflection and stresses incorporating different thickness schemes of sandwich shells are presented in the form of figures. To validate the results, a functionally graded shell without sandwich arrangement is considered. Since no results are available on static analysis of FGMSS, the present 2D model based on the finite element method might be helpful in assessing the applicability of other analytical and numerical models in this area in the future.     

四边固支功能梯度板中波的传播

摘要：考虑剪切变形和转动惯性的影响,采用一阶剪切变形板理论和小应变的应变－位移关系,利用Hamilton原理推得运动控制方程,并应用特征值方程得到频散方程。给出了波在功能梯度板中传播的频散,相速度和群速度随波数变化的曲线,分析了不同的功能梯度材料指数对波传播的影响规律。     

Dynamic response of functionally graded beams in a thermal environment under a moving load

The dynamic response of functionally graded (FG) beams in thermal environment subjected to moving load is investigated based on the first-order shear deformation theory (FSDT). The initial thermal stresses are determined by solving the thermoelastic equilibrium equations. The finite element method (FEM) is adopted to develop a solution procedure for FG beams with general loading and boundary conditions. The convergence behavior and accuracy of the method are shown through the different numerical examples. Finally, the influences of temperature rise, material graded index, moving load velocity, and boundary conditions on the dynamic behavior of FG beams in thermal environment is presented.     

受机械力作用金属／陶瓷功能梯度板的非线性振动模型

研究机械力作用下金属，陶瓷功能梯度薄板的建模问题。应用弹性理论和Galerkin方法建立小挠度金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。     

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。      

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.     

Imperfection sensitivity in the nonlinear vibration of initially stresses functionally graded plates

In this paper, the nonlinear partial differential equations of nonlinear vibration for an imperfect functionally graded plate (FGP) in a general state of arbitrary initial stresses are presented. The derived equations include the effects of initial stresses and initial imperfections size. The material properties of a FGP are graded continuously in the direction of thickness. The variation of the properties follows a simple power-law distribution in terms of the volume fractions of the constituents. Using these derived governing equations, the nonlinear vibration of initially stressed FGPs with geometric imperfection was studied. The present approach employed a perturbation technique, the Galerkin method and the Runge–Kutta method. The perturbation technique was used to derive the nonlinear governing equations. The motion of imperfect FGPs was obtained by performing the Galerkin method and then solved by the Runge–Kutta method. Numerical solutions are presented for the performances of perfect and imperfect FGPs. The nonlinear vibration of a simply supported ceramic/metal FGP was solved. It is found that the initial stress, geometric imperfection and volume fraction index greatly change the behavior of nonlinear vibration.     

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.     

Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads

Based on Reddy's higher-order shear deformation plate theory, this article presents an analysis of the nonlinear dynamic response and vibration of imperfect functionally graded material (FGM) thick plates subjected to blast and thermal loads resting on elastic foundations. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Numerical results for the dynamic response and vibration of the FGM plates with two cases of boundary conditions are obtained by the Galerkin method and fourth-order Runge–Kutta method. The results show the effects of geometrical parameters, material properties, imperfections, temperature increment, elastic foundations, and boundary conditions on the nonlinear dynamic response and vibration of FGM plates.     

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。     

Stability analysis of FGM sandwich plates by using variable-kinematics Ritz models

Abstract

A linearized buckling analysis of functionally graded material (FGM) isotropic and sandwich plates is carried out by virtue of the Hierarchical Trigonometric Ritz Formulation (HTRF). Quasi-3D Ritz models based on equivalent single layer (ESL) and zig zag (ZZ) plate theories are developed within the framework of the Carrera Unified Formulation (CUF). Several in-plane loading conditions accounting for axial, biaxial, and shear loadings are taken into account. Parametric studies are carried out in order to evaluate the effects of significant parameters, such as volume fraction index, length-to-thickness ratio, sandwich plate type, and loading type, on the critical buckling loads.     

热流温度场下功能梯度板的热问题研究

功能梯度材料因其内部组分沿着空间位置连续变化,能有效缓解热应力集中等现象,在高超音速飞行器的热防护系统设计中具有良好的应用前景.以金属-陶瓷功能梯度板为研究对象,探讨在不同热环境下功能梯度板热传导、热变形和热应力的变化规律.首先,基于功能梯度材料的幂律分布模型,分析了线性温度场、正弦温度场、热流温度场和非线性温度场四种...