Hyperbolic Heat Conduction in a Functionally Graded Hollow Sphere

Non-Fourier hyperbolic heat conduction in a heterogeneous sphere is investigated in this article. Except for the thermal relaxation time, which is assumed to be constant, all other material properties vary continuously within the sphere in the radial direction following a power law. Boundary conditions of the sphere are assumed to be spherically symmetric, leading to a one-dimensional heat conduction problem. The problem is solved analytically in the Laplace domain, and the final results in the time domain are obtained using numerical inversion of the Laplace transform. The transient responses of temperature and heat flux are investigated for different non-homogeneity parameters and normalized thermal relaxation constants. The current results for the specific case of a homogeneous sphere are validated by results available in the literature.     

Heat Conduction Analysis of Nonhomogeneous Functionally Graded Three-Layer Media

Functionally graded material (FGM) is a particulate composite with continuously changing its thermal and mechanical properties in order to raise the bonding strength in the discrete composite made from different phases of material constituents. Furthermore, FGM is a potent tool to create an intermediate layer in metal–ceramic composites to avoid the properties discontinuities and reduce, thereby, the residual stresses. For the nonhomogeneous problem, the mathematical derivation is much complicated than the homogeneous case since the material properties vary with coordinate. To analyze the problem, the Fourier transform is applied and the general solution in transform domain is obtained. The inverse Fourier transform is performed to get the results in physical domain for temperature and heat fluxes. Numerical results for the full-field distributions of temperature and heat fluxes with different functionally graded parameters are presented. The continuous characteristics of the temperature and heat flux along the interface are emphasized and some interesting phenomena are presented in this study. The results show that all the fields (temperature and heat fluxes) are continuous at the interface if the conductivities are continuous at the interface. Moreover, the first derivatives of temperature and heat flux qy are continuous at the interface.     

功能梯度材料热应力研究进展

梯度功能材料的热应力问题贯穿梯度功能材料设计、制备、性能评价及应用整个研究领域,其中,热传导问题是热应力研究基础。介绍了梯度功能材料的概念及热传导和热应力问题的研究背景,重点分析了梯度功能材料热传导和热应力问题在数学模型、物性参数模型、解析方法、数值方法等方面的国内外研究进展,并展望进一步研究方向。     

对流换热边界下梯度功能材料板瞬态热传导有限元分析

用有限元法与有限差分法相结合的方法 ,对处在对流换热边界条件下的梯度功能材料板的瞬态热传导问题进行了分析 ,并且通过由ZrO2 和Ti 6A1 4V组成的梯度功能材料板对本方法的正确性进行了检验 ,最后给出了对流换热边界下的瞬态温度场分布。数值计算结果表明 :材料组分的分布形状系数M、环境介质温度和对流换热系数的变化对梯度功能材料板的瞬态温度场分布均有明显的影响。本文结果为梯度功能材料的优化设计和进一步的热应力分析提供了理论计算依据。     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

Calculation of the Composition Profile of a Functionally Graded Material Produced by Co-sedimentation

A sedimentation method is proposed to fabricate functionally graded materials (FGMs) with the continuous and smooth variations of composition .The relations between the compositional distribution of deposited body and the powder characteristics of raw materials as well as settling parameters are dervied,Subsequently ,the mathematical model of forming FGM based on the co-sedimentation has been established,At last,numerical simulations are conducted to explore the effects of the particle sizes of raw materilas and suspension height on the compositional distribution of final products.     

Thermal Buckling Analysis of Piezoelectric Functionally Graded Plates with Temperature-Dependent Properties

In this study, the thermal buckling analysis of hybrid laminated plates made of two-layered functionally graded materials (FGMs) that are integrated with surface-bonded piezoelectric actuators referred to as (P/FGM)_s are investigated. Material properties for both substrate FGM layers and piezoelectric layers are temperature-dependent. Uniform temperature rise as a thermal load and constant applied actuator voltage are considered for this analysis. By definition of four new analytic functions, the five coupled governing stability equations, which are derived based on the first-order shear deformation plate theory, are converted into fourth-order and second-order decoupled partial differential equations (PDEs). Considering a Levy-type solution, these two PDEs are reduced to two ordinary differential equations. One of these equations is solved using an accurate analytical solution, which is named as power series Frobenius method. The effects of parameters, such as the plate aspect ratio, ratio of piezoelectric layer thickness to thickness of FGM layer, gradient index, actuator voltage, and the temperature dependency on the critical buckling temperature difference, are illustrated and explained. The critical buckling temperatures of (P/FGM)_s with six various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Finite Element Multi-mode Approach to Thermal Postbuckling of Functionally Graded Plates

Postbuckling analysis of functionally graded ceramic-metal plates under temperature field is presented using finite element multi-mode method. The three-node triangular element based on the Mindlin plate theory is employed to account for the transverse shear strains, and the von-Karman nonlinear strain-displacement relation is utilized considering the geometric nonlinearity. The effective material properties are assumed to vary through the thickness direction according to the power law distribution of the volume fraction of constituents. The temperature distribution along the thickness is determined by one dimensional Fourier equations of heat conduction. The buckling mode shape solved from eigen-buckling analysis is adopted as the assumed mode function to reduce the degrees of freedom of nonlinear postbuckling equilibrium equations. The postbuckling response is obtained by solving the nonlinear equilibrium equations, and compared with the Newton- Raphson numerical results. The effects of boundary conditions, material gradient index and temperature distribution on postbuckling behavior are examined.     

功能梯度材料制备过程影响因素的数值研究

本文用数值计算的方法对重力浇铸下Al—Si／SiC颗粒系统(合金基质的功能梯度材料)凝固过程进行了研究，分析了凝固条件及各种不同参数对铸件中颗粒和溶质浓度分布的影响。结果表明，凝固件中颗粒体积分数分布都大致可分为三个区域：靠近底部的颗粒堆积区，靠近顶部的颗粒体积分数减小或近似为零区，及中部附近颗粒体积分数近似保持不变区。浇铸的初始温度，颗粒初始体积分数，颗粒直径和冷却速率等参数对颗粒和溶质浓度分布有很大影响。     

Analyses of Circular Magnetoelectroelastic Plates with Functionally Graded Material Properties

A meshless method based on the local Petrov–Galerkin approach is proposed for plate bending analysis with material containing functionally graded magnetoelectroelastic properties. Material properties are considered to be continuously varying along the plate thickness. Axial symmetry of geometry and boundary conditions for a circular plate reduces the original 3D boundary value problem into a 2D problem in axial cross section. Both stationary and transient dynamic conditions for a pure mechanical load are considered in this article. The local weak formulation is employed on circular subdomains in the axial cross section. Subdomains surrounding nodes are randomly spread over the analyzed domain. The test functions are taken as unit step functions in derivation of the local integral equations (LIEs). The moving least-squares (MLS) method is adopted for the approximation of the physical quantities in the LIEs. After performing the spatial integrations, one obtains a system of ordinary differential equations for certain nodal unknowns. That system is solved numerically by the Houbolt finite-difference scheme as a time-stepping method.     

Bending,Free Vibration and Buckling Analysis of Functionally Graded Plates via Wavelet Finite Element Method

Following previous work, a wavelet finite element method is developed for bending, free vibration and buckling analysis of functionally graded (FG) plates based on Mindlin plate theory. The functionally graded material (FGM) properties are assumed to vary smoothly and continuously throughout the thickness of plate according to power law distribution of volume fraction of constituents. This article adopts scaling functions of two-dimensional tensor product BSWI to form shape functions. Then two-dimensional FGM BSWI element is constructed based on Mindlin plate theory by means of two-dimensional tensor product BSWI. The proposed two-dimensional FGM BSWI element possesses the advantages of high convergence, high accuracy and reliability with fewer degrees of freedoms on account of the excellent approximation property of BSWI. Numerical examples concerning various length-to-thickness ratios, volume fraction indexes, aspect ratios and boundary conditions are carried out for bending, free vibration and buckling problems of FG plates. These comparison examples demonstrate the accuracy and reliability of the proposed WFEM method comparing with the exact and referential solutions available in literatures.     

Three-Dimensional Solutions of Functionally Graded Piezo-Thermo-Elastic Shells and Plates Using a Modified Pagano Method

A modified Pagano method is developed for the three-dimensional (3D) coupled analysis of simply-supported, doubly curved functionally graded (FG) piezo-thermo-elastic shells under thermal loads. Four different loading conditions, applied on the lateral surfaces of the shells, are considered. The material properties of FG shells are regarded as heterogeneous through the thickness coordinate, and then specified to obey an exponent-law dependent on this. The Pagano method, conventionally used for the analysis of multilayered composite elastic plates/shells, is modified to be feasible for the present analysis of FG piezo-thermo-elastic plates/shells. The modifications include that a displacement-based formulation is replaced by a mixed formulation, a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions, a successive approximation (SA) method is adopted and introduced in the present analysis, and the propagator matrix method is developed for the heat conduction analysis and the coupled piezo-thermo-elastic analysis of the FG shells. The influence of the material-property gradient index on the field variables, induced in the FG piezo-thermo-elastic shells and plates under the thermal load, is studied.     

An Application of DHW Algorithm for the Solution of Inverse Heat Conduction Problem

A damped heat wave (DHW) algorithm is applied for the temperature distribution calculation in a solution of a linear inverse heat conduction problem (IHCP). A nonlinear least squares algorithm is used for calculation of the unknown boundary heat flux history in a one-dimensional medium. The solution is based on the assumption that the temperature measurements are available, at least, at one point of the medium over the whole time domain. Sample calculations, for a comparison between exact heat sources and estimated ones, are made to confirm the validity of the proposed method. The close agreement between the exact and estimated values calculated for both exact and noisy data shows the potential of the proposed method for finding a relatively accurate heat source distribution in a one-dimensional homogeneous finite medium. The proposed method of solving inverse heat conduction problems is very simple and easy to implement.Paper presented at the Seventeenth European Conference on Thermophysical Properties, September 5–8, 2005, Bratislava, Slovak Republic.M. L?ffler: Deceased     

A Meshless Local Petrov-Galerkin Method for the Analysis of Cracks in the Isotropic Functionally Graded Material

A meshless local Petrov-Galerkin method (MLPG) [[Atluri and Zhu (1998)] for the analysis of cracks in isotropic functionally graded materials is presented. The meshless method uses the moving least squares (MLS) to approximate the field unknowns. The shape function has not the Kronecker Delta properties for the trial-function-interpolation, and a direct interpolation method is adopted to impose essential boundary conditions. The MLPG method does not involve any domain and singular integrals to generate the global effective stiffness matrix if body force is ignored; it only involves a regular boundary integral. The material properties are smooth functions of spatial coordinates and two interaction integrals [Rao and Rahman (2003a,b)] are used for the fracture analysis. Two numerical examples including both mode-I and mixed-mode problems are presented to calculated the stress intensity factors (SIFs) by the proposed method. Example problems in functionally graded materials are presented and compared with available reference solutions. A good agreement obtained show that the proposed method possesses no numerical difficulties.     

Fabrication of ZrO2/Mo-Si/Ni Functionally Graded Material by Dip-Coating

A slurry dip-coating technique was developed for fabrication of ZrO2/Mo-Si/Ni functionally graded material (FGM) on the stainless steel substrate,The rheological behavior of ZrO2-Ni-ethanol slurry was characterized by viscosity test,The amount of polyvinyl butyral (PVB) additives,which ser5ved as the dispersant and binder in ZrO2-Ni-ethanol slurry ,was optimized,The results showed that the characters of mixed slurries,with added 9 vol,pct (relatively to total powders )MoSi2 powders prepared by mechanical alloying changed little ,The stainless steel substrate was coated several times by dipping in the slurries,and followed by drying in air every dipping,After debinding in Ar in graphite die,the coated FGM plate was finally hot pressed at 1300 ℃ for 1 h under the pressure of 5 MPa in Ar in the same die,Microstructural observations of the sintered FGM specimens revealed that the graded layers were formed on the stainless steel substrate ,in which no cracks were observed.     

Buckling and Postbuckling Behavior of Functionally Graded Nanotube-Reinforced Composite Plates in Thermal Environments

This paper investigates the buckling and postbuckling of simply supported, nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. The nanocomposite plates are assumed to be functionally graded in the thickness direction using single-walled carbon nanotubes (SWCNTs) serving as reinforcements and the plates' effective material properties are estimated through a micromechanical model. The higher order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity is used to model the composite plates and a two-step perturbation technique is performed to determine the buckling loads and postbuckling equilibrium paths. Numerical results for perfect and imperfect, geometrically mid-plane symmetric functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC plate, which is a special case in the present study, are compared with those of the FG-CNTRC plate. The results show that the buckling loads as well as postbuckling strength of the plate can be significantly increased as a result of a functionally graded nanotube reinforcement. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC plates.     

Eigen-vibrations of Plates made of Functionally Graded Material

Within the framework of the direct approach to the plate theory we consider natural oscillations of plates made of functionally graded materials taking into account both the rotatory inertia and the transverse shear stiffness. It is shown that in some cases the results based on the direct approach differ significantly from the classical estimates. The reason for this is the non-classical computation of the transverse shear stiffness.     

离心铸造SiCp／A356功能梯度材料的组织结构与耐磨性

采用离心铸造法制备了ＳｉＣｐ／Ａ３５６功能梯度材料并研究了其组织结构及耐磨性。结果表明，离心铸造ＳｉＣｐ／Ａ３５６梯度材料组织致密，ＳｉＣ粒子在材料中呈梯度分布；在离心力场的作用下，消除了ＳｉＣ粒子团聚常来的缺陷，使ＳｉＣ粒子与合金基体紧密结合，充分发挥了ＳｉＣ粒子的优良性能，从而较大地提高了材料的耐磨性。     

A Differential Quadrature Method for Multi-Dimensional Inverse Heat Conduction Problem of Heat Source

In this paper, we employ the differential quadrature method (DQM) to tackle the inverse heat conduction problem (IHCP) of heat source. These advantages of this numerical approach are that no a priori presumption is made on the functional form of the estimates, and that evaluated heat source can be obtained directly in the calculation process. Seven examples show the effectiveness and accuracy of our algorism in providing excellent estimates of unknown heat source from the given data. We find that the proposed scheme is applicable to the IHCP of heat source. Even though the noise is added to the exact temperature, the DQM is still robust against disturbance.     

A Method of Material Optimization of Cementless Stem Through Functionally Graded Material

Ideally, a bone implant should be such that it exhibits an identical response to loading as real bone and is also biocompatible with existing tissue. A stiff stem, which is usually made of titanium, shields the proximal bone from mechanical loading (stress shielding). On the other hand, decreasing the stem stiffness increases the proximal interface shear stress and the risk of proximal interface failure. Therefore the purpose of this study is to solve these conflicting requirements in order to have more uniform interface shear stress distribution and less stress shielding through the concept of functionally graded material (FGM). FGM is a kind of advanced composite materials, which changes its composition and structure gradually over one or two directions of its volume, resulting in corresponding changes in the properties of the material. This study is divided into two parts; in the first part, the finite element analysis and optimization technique are used to design the stem as one-dimensional FGM, while in the second part, the stem is designed as two-dimensional functionally graded material. The aim of both designs is to overcome the above mentioned problems. In the case of part one (one-dimensional FGM), the gradation of elastic modulus is changed along the vertical direction (model 1) and along the horizontal direction (model 2), in order to find the optimal gradation direction. It is found that the optimal design is to change the elastic modulus gradually from 110 GPa (Hydroxyapatite) at the top of the stem to 1GPa (Collagen) at the bottom (model 1). This optimal gradation decreases stress shielding by 83%, while reduces the maximum interface shear stress by 32% compared to homogenous titanium stem. However, in the second part (two-dimensional FGM, model 3) the materials of optimal design are found to be hydroxyapatite, Bioglass, and collagen. This design leads to the same stress shielding reduction as in model 1, while at the same time, the maximum interface shear stress is reduced by 45% and 63% compared to the optimal one-dimensional FGM design and homogenous titanium stem, respectively.