Stochastic analysis of a thermoelastic problem in functionally graded plates with uncertain material properties

The statistics (i.e., mean and variance) of temperature and thermal stress are analytically obtained in functionally graded material (FGM) plates with uncertainties in the thermal conductivity and coefficient of linear thermal expansion. These FGM plates are assumed to have arbitrary nonhomogeneous thermal and mechanical properties through the entire thickness of plate and are subjected to deterministic convective heating. The stochastic temperature and thermal stress fields are analysed by assuming the FGM plate is multilayered with distinct, random thermal conductivity and coefficient of linear thermal expansion in each layer. Vodicka’s method, which is a type of integral transform method, and a perturbation method are employed to obtain the analytical solutions for the statistics. The autocorrelation coefficients of each random property and cross-correlation coefficients between different random properties are expressed in exponential function forms as a non-homogeneous Markov random field of discrete space. Numerical calculations are performed for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304), which have the largest dispersion of the random properties at the place where the volume fractions of the two constituent materials are both 0.5. The effects of the spatial change in material composition, thermal boundary condition and correlation coefficients on the standard deviations of the temperature and thermal stress are discussed.     

Information Measures for Generalized Order Statistics and Their Concomitants under General Framework from Huang-Kotz FGM Bivariate Distribution

In this paper, we study the concomitants of dual generalized order statistics (and consequently generalized order statistics) when the parameters γ1,…,γn are assumed to be pairwise different from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution. Some useful recurrence relations between single and product moments of concomitants are obtained. Moreover, Shannon’s entropy and the Fisher information number measures are derived. Finally, these measures are extensively studied for some well-known distributions such as exponential, Pareto and power distributions. The main motivation of the study of the concomitants of generalized order statistics (as an important practical kind to order the bivariate data) under this general framework is to enable researchers in different fields of statistics to use some of the important models contained in these generalized order statistics only under this general framework. These extended models are frequently used in the reliability theory, such as the progressive type-II censored order statistics.     

含孔隙变厚度FG圆板的湿热力学响应

功能梯度材料(functionally graded materials, FGM)是组份含量按特定方向连续变化的非均匀复合材料,可有效解决传统复合材料组份之间结合能力弱和不同组份性能难以协调等问题,达到诸如缓和应力集中和优化应力分布等效果,使整体材料在保持细观结构完整性的同时充分发挥各组份材料的性能优势.由于制备技术等原因或出于特殊功能的需要,微孔或孔隙是各类型FGM中的常见缺陷.从细观结构上看,多孔FGM中的孔隙包含了单一组份内的材料孔隙和组份微粒间的结构间隙,这些孔隙将对FGM的力学性能,尤其是在湿热环境下的力学行为产生影响.本文考虑FGM中的两类细观孔隙(材料孔隙和结构孔隙),提出了令各类孔隙依赖于各自组份变化,再线性叠加得到的整体孔隙计算式.考虑组份材料和孔隙填充物(液相水和水蒸气)性质的温度相关性,建立了湿热相关FGM材料模型.针对厚度沿径向变化的旋转圆板结构,应用该FGM材料模型,推导了圆板的非线性稳态湿热控制方程及考虑湿热弹性本构的位移控制方程,采用微分求积法(differential quadrature method,DQM),获得了圆板的湿热场、位移场和应力分布.在数值算例中,利用退化模型的解析解对本文的数值计算方法进行了验证,继而通过改变各关键参数,讨论了两类孔隙率、梯度指数和圆板厚度变化对含孔隙FGM变厚度旋转圆板湿热力学响应的影响规律.      

Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories

The free vibration of functionally graded material （FGM） beams is studied based on both the classical and the first-order shear deformation beam theories. The equations of motion for the FGM beams are derived by considering the shear deforma- tion and the axial, transversal, rotational, and axial-rotational coupling inertia forces on the assumption that the material properties vary arbitrarily in the thickness direction. By using the numerical shooting method to solve the eigenvalue problem of the coupled ordinary differential equations with different boundary conditions, the natural frequen- cies of the FGM Timoshenko beams are obtained numerically. In a special case of the classical beam theory, a proportional transformation between the natural frequencies of the FGM and the reference homogenous beams is obtained by using the mathematical similarity between the mathematical formulations. This formula provides a simple and useful approach to evaluate the natural frequencies of the FGM beams without dealing with the tension-bending coupling problem. Approximately, this analogous transition can also be extended to predict the frequencies of the FGM Timoshenko beams. The numerical results obtained by the shooting method and those obtained by the analogous transformation are presented to show the effects of the material gradient, the slenderness ratio, and the boundary conditions on the natural frequencies in detail.     

梯度功能压电悬臂梁的一组基本解及其应用

采用应力函数解法,研究了弹性参数和体积力同时呈梯度变化时压电材料悬臂粱的力-电响应,得到了应力函数和电位移函数的解析表达式及梯度功能压电悬臂梁的一组基本解．作为一种应用形式,给出了梯度功能压电执行器的尖端位移和制动力的确定方法、此外,利用该基本解,可以方便地确定悬臂梁在多种不同典型荷载单独或联合作用下的解答。     

Anti-plane elastodynamic analysis of cracked graded orthotropic layers with viscous damping

Stress analysis is carried out in a graded orthotropic layer containing a screw dislocation undergoing time-harmonic deformation. Energy dissipation in the layer is modeled by viscous damping. The stress fields are Cauchy singular at the location of dislocation. The dislocation solution is utilized to derive integral equations for multiple interacting cracks with any location and orientation in the layer. These equations are solved numerically thereby obtaining the dislocation density function on the crack surfaces and stress intensity factors of cracks. The dependencies of stress intensity factors of cracks on the excitation frequency of applied traction and material properties of the layer are investigated. The analysis allows the determination of natural frequencies of a cracked layer. Furthermore, the interactions of two cracks having various configurations are studied.     

Interaction of a plane progressive sound wave with a functionally graded spherical shell

An exact analysis is carried out to study interaction of a time-harmonic plane progressive sound field with a radially inhomogeneous thick-walled elastic isotropic spherical shell suspended in and filled with compressible ideal fluid mediums. Using the laminated approximation method, a modal state equation with variable coefficients is set up in terms of appropriate displacement and stress functions and their spherical harmonics. Taylor’s expansion theorem is then employed to obtain the solution to the modal state equation ultimately leading to calculation of a global transfer matrix. Numerical example is given for a water-submerged/air-filled Aluminum/Zirconia elastic spherical sandwich shell containing a functionally graded interlayer and subjected to an incident progressive plane sound wave. The mechanical properties of the interlayer are assumed to vary smoothly and continuously across the thickness with the change of volume concentration of its constituents. The effect of incident wave frequency, thickness and compositional gradient of the interlayer on the form function amplitude and the average radiation force acting on the composite shell are examined. Limiting cases are considered and fair agreements with well-known solutions are established.     

梯度材料平板弯拉耦合力学的精确化支配方程

基于非均匀材料弹性力学理论,采用算子谱分解和算子代数方法,对梯度材料平板结构弯曲与拉伸问题进行了研究.首次给出了指数梯度材料平板弯曲与拉伸的力学方程.研究结果表明:与各向同性平板结构弯曲和拉伸问题不同,在功能梯度平板中描述弯曲应力状态与描述拉伸应力状态的广义位移函数以及剪切函数都是耦合的.没有采用工程假设,推导得到的梯度材料平板结构力学方程是精确化的.通过分析可以认识和理解,分别对应于弯曲状态与拉伸状态的应力场耦合机理以及力学响应的构成等.给出的方程及其分析过程可望能够用于类平板形式热防护材料结构的应力分析与强度设计,推进热防护材料结构的轻型化.     

Reduction of thermal stresses by developing two-dimensional functionally graded materials

Modern aerospace shuttles and craft are subjected to super high temperatures, that have variation in two or three directions, which need to introduce new materials that can stand with such applications. Therefore, in the present work a two-dimensional functionally graded materials, 2D-FGM, are introduced to withstand super high temperatures and to give more reduction in thermal stresses. The suitable functions that can represent volume fractions of the introduced 2D-FGM are proposed. Then the rules of mixture of the 2D-FGM are derived based on the volume fractions of the 2D-FGM and the rules of mixture of the conventional FGM. The introduced volume fractions and rules of mixture for 2D-FGM were used to calculate the thermal stresses in 2D-FGM plate. Comparison between 2D-FGM and conventional FGM was carried out and showed that 2D-FGM has high capability to reduce thermal stresses than the conventional FGM.