Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials(FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier's equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement,strains, and stresses are determined by the exact solution to Navier's equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.     

A Pressurized Functionally Graded Hollow Cylinder with Arbitrarily Varying Material Properties

The elastic analysis of a pressurized functionally graded material (FGM) annulus or tube is made in this paper. Different from existing studies, this study deals with an axisymmetrical FGM hollow cylinder or disk with arbitrarily varying material properties. A simple and efficient approach is suggested, which reduces the associated problem to solving a Fredholm integral equation. The resulting equation is approximately solved by expanding the solution as series of Legendre polynomials. The stresses and displacements can be represented in terms of the solution to the equation. For radius-dependent Young’s modulus, numerical results of the distribution of the radial and circumferential stresses are presented graphically. Our results indicate that change in the gradient of the FGM tube does not produce a substantial variation of the radial stress, but strongly affects the distribution of the hoop stress. In particular, the hoop stress may reach its maximum at an internal position or at the outer surface when the tube is internally pressurized. The results obtained are helpful in designing FGM cylindrical vessels to prevent failure.      

Two-dimensional thermoelastic contact problem of functionally graded materials involving frictional heating

The two-dimensional thermoelastic sliding frictional contact of functionally graded material (FGM) coated half-plane under the plane strain deformation is investigated in this paper. A rigid punch is sliding over the surface of the FGM coating with a constant velocity. Frictional heating, with its value proportional to contact pressure, friction coefficient and sliding velocity, is generated at the interface between the punch and the FGM coating. The material properties of the coating vary exponentially along the thickness direction. In order to solve the heat conduction equation analytically, the homogeneous multi-layered model is adopted for treating the graded thermal diffusivity coefficient with other thermomechanical properties being kept as the given exponential forms. The transfer matrix method and Fourier integral transform technique are employed to convert the problem into a Cauchy singular integral equation which is then solved numerically to obtain the unknown contact pressure and the in-plane component of the surface stresses. The effects of the gradient index, Peclet number and friction coefficient on the thermoelastic contact characteristics are discussed in detail. Numerical results show that the distribution of the contact stress can be altered and therefore the thermoelastic contact damage can be modified by adjusting the gradient index, Peclet number and friction coefficient.     

Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment

Free vibration analysis of functionally graded (FG) thin-to-moderately thick annular plates subjected to thermal environment and supported on two-parameter elastic foundation is investigated. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle based on the first order shear deformation theory (FSDT). The initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. Differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with the available solutions in the literature for isotropic and FG circular and annular plates. The effects of the temperature rise, elastic foundation coefficients, the material graded index and different geometrical parameters on the frequency parameters of the FG annular plates are investigated. The new results can be used as benchmark solutions for future researches.     

Two-dimensional thermoelastic analysis of a functionally graded cylindrical panel due to nonuniform heat supply

This paper is concerned with the theoretical treatment of the steady-state thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply in the circumferential direction. The thermal and thermoelastic constants of the cylindrical panel are expressed as power functions of the radial coordinate. We obtain the exact solution for the two-dimensional temperature change in a steady state, and thermal stresses of a simple supported cylindrical panel under the state of plane strain. Some numerical results are shown in figures and tables. Furthermore, the influence of the nonhomogeneity of the material, the radius ratio and the span angle upon the temperature change, displacements and stresses is investigated.     

功能梯度矩形厚板的三维热弹性分析

直接从三维热弹性力学基本方程出发，通过引入两个位移函数和两个应力函数，导出了一个二阶的齐次状态方程和一个四阶的非齐次状态方程。分析中采用了层合近似模型，即将板划分成厚度足够小的若干薄层，从面可将每一层内的材料常数近似为常数。给出了任意厚度的四边简支横观各向同性功能梯度矩形板的热弹性分析，特别当板较薄时，与薄板理论进行了数值对比，发现两者结果吻合很好。最后研究了材料梯度指标对热弹性场的影响，结果显示梯度指标对热应力和位移都有着显著的作用：在不同的区间，梯度指标对它们有不同的影响；并且在同一区间，梯度指标对两者的影响程度也有所不同。     

Deformations and stresses in annular disks made of functionally graded materials subjected to internal and/or external pressure

In this paper, an analytical solution is developed to determine deformations and stresses in circular disks made of functionally graded materials subjected to internal and/or external pressure. Taking mechanical properties of the materials of circular disks to be linear variations, the governing equation is derived from basic equations of axisymmetric, plane stress problems in elasticity. By transforming the governing equation into a hypergeometric equation, an accurate analytical solution of deformations and stresses in circular disks is obtained. The comparison with the numerical solution indicates that both approaches give very agreeable results, indicating correctness of the proposed analytical solution. The obtained analytical solution is employed to determine the radial displacement and stresses in circular disks subjected to external pressure, internal pressure, and internal and external pressure, respectively. How the radius ratio of circular disks affects deformations and stresses is also investigated.     

Analysis for the thermoelastic bending of a functionally graded material cylindrical shell

An analytic study for thermoelastic bending of a functionally graded material (FGM) cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived out. An arbitrary material property of the FGM cylindrical shell is assumed to vary through the thickness of the cylindrical shell, and exact solution of the problem is obtained by using an analytic method. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load and the power law exponent on the deformation of the FGM cylindrical shell are analyzed and discussed.     

功能梯度压电材料圆环在位移和反对称电势边值条件下的解析解

采用了在径向极化情况下横观各向同性的线性本构关系,考虑了材料性质沿径向的梯度分布,对功能梯度压电材料圆环在给定的位移和电势边界条件下,导出了问题的一般解．推导了外壁固定、接地,内壁沿垂向有一微小位移、电势为反对称分布问题的解析解,并计算了该问题在位移和电势作用情况下的位移、电势在不同梯度分布时的数值结果．     

Coupled thermoelasticity of functionally graded cylindrical shells

The coupled thermoelastic response of a functionally graded circular cylindrical shell is studied. The coupled thermoelastic and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. A second-order shear deformation shell theory that accounts for the transverse shear strains and rotations is considered. Including the thermo-mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain are used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The results are validated with the known data in the literature.     

On an inverse problem for inhomogeneous thermoelastic rod

In recent years, different fields of engineering have been increasingly incorporating functionally graded materials with variable physical properties that significantly improve a quality of elements of designs. The efficiency of practical application of thermoelastic inhomogeneous materials depends on knowledge of exact laws of heterogeneity, and to define them it is necessary to solve coefficient inverse problems of thermoelasticity.In the present research a scheme of solving the inverse problem for an inhomogeneous thermoelastic rod is presented. Two statements of the inverse problem are considered: in the Laplace transform space and in the actual space. The direct problem solving is reduced to a system of the Fredholm integral equations of the 2nd kind in the Laplace transform space and an inversion of the solutions obtained on the basis of the theory of residues. The inverse problem solving is reduced to an iterative procedure, at its each step it is necessary to solve the Fredholm integral equation of the 1st kind; to solve it the Tikhonov method is used. Specific examples of a reconstruction of variable characteristics required are given.     

热环境中黏弹性功能梯度材料及其结构的蠕变

基于经典的对应原理, 将 Mori-Tanaka 方法等细观力学结果推广于定常温度环境下的黏弹性情形. 根据泊松比与时间呈弱相关的特点, 给出 Laplace 象空间中功能梯度材料的松弛模量和热膨胀系数, 并直接建立耦合热应变的多维黏弹性本构关系. 在此基础上, 求解黏弹性功能梯度圆柱薄壳在热环境中的轴对称弯曲蠕变变形问题. 考虑材料热物参数的温度相关性, 首先确定稳态温度场, 导出相空间中轴对称弯曲变形的解析解, 采用数值反演得到蠕变变形. 算例表明, 蠕变初期, 热环境的影响明显, 随着时间增加, 热应力松弛, 影响逐渐消失. 当圆柱薄壳受轴压时, 相比于两端固支, 两端简支的端部变形更加明显. 通过圆柱薄壳的轴对称弯曲求解, 给出体积含量呈任意分布的黏弹性功能梯度结构在热机载荷下的蠕变分析途径.      

功能梯度旋转圆板的热弹性固有振动

在考虑热因素及旋转运动条件下,针对金属－陶瓷功能梯度圆板的固有振动问题进行研究．给出随温度变化且材料组分沿厚度方向按幂律分布的材料物性参数,依据热弹性理论得到圆板的能量关系式．基于哈密顿原理建立旋转金属－陶瓷功能梯度圆板热弹性动力学方程．采用伽辽金法得到边界约束下圆板的自由振动方程,确定了静挠度及固有振动频率．基于数值计算,得到系统固有频率值随体积分数指数、转速和温度等参量的变化曲线,讨论了静挠度变化规律及动力系统的奇点稳定性问题．结果表明,固有频率随体积分数指数、材料表面温度以及转速的增加而减小．     

Thermoelastic contact instability of a functionally graded layer and a homogeneous half-plane

The conductive heat transfer between two elastic bodies in the static contact can cause the system to be unstable due to the interaction between the thermoelastic distortion and pressure-dependent thermal contact resistance. This paper investigates the thermoelastic contact instability of a functionally graded material (FGM) layer and a homogeneous half-plane using the perturbation method. The FGM layer and half-plane are exposed to a uniform heat flux and are pressed together by a uniform pressure. The material properties of the FGM layer vary exponentially along the thickness direction. The characteristic equation governing the thermoelastic stability behavior is obtained to determine the stability boundary. The effects of the gradient index, layer thickness and material combination on the critical heat flux are discussed in detail through a parametric study. Results indicate that the thermoelastic stability behavior can be modified by adjusting the gradient index of the FGM layer.     

轴对称自由振动功能梯度材料微圆板中的热弹性阻尼

基于经典弹性薄板理论和单向耦合热传导理论,研究了材料性质沿厚度连续变化的功能梯度微圆板的热弹性阻尼特性．首先,考虑热力耦合效应,建立了功能梯度微圆板轴对称横向自由振动微分方程．然后,忽略温度梯度在面内的变化,建立了单向耦合变系数一维热传导方程．采用分层均匀化近似方法,将变系数热传导方程转化为一系列常系数的微分方程,利用上下表面的热边界条件和层间连续性条件获得了微圆板温度场解析解．将所得温度场代入微圆板的自由振动微分方程,得到了包含热弹性阻尼的复频率,从而获得了反映热弹性阻尼水平的逆品质因子．最后,针对材料性质沿板厚按幂函数变化的陶瓷-金属功能梯度微圆板,定量地分析材料梯度指数、几何尺寸、边界条件、温度环境等对微圆板热弹性阻尼的影响．     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

功能梯度材料涂层半空间的轴对称光滑接触问题

求解了功能梯度材料涂层半空间的轴对称光滑接触问题,其中梯度层剪切模量按照线性变化,利用Hankel积分变换方法求解微分方程,将问题化为具有Cauchy型奇异核的积分方程.数值方法求解表明:功能梯度材料涂层半空间在刚性柱形压头和球形压头作用下,接触表面分布应力,接触半径以及最大压痕受材料梯度效应的影响较大.     

Application and analysis of functionally graded piezoelectrical rotating cylinder as mechanical sensor subjected to pressure and thermal loads

The exact thermoelastic analysis of a functionally graded piezoelectrical (FGP) rotating cylinder is investigated analytically. The cylinder is subjected to a combination of electrical, thermal, and mechanical loads simultaneously. The structure is a simplified model of a rotational sensor or actuator. The basic governing differential equation of the system is obtained by using the energy method. A novel term, named as the additional energy, is introduced to exact the evaluation of the energy functional. The solution to the governing differential equation is presented for two types of boundary conditions including free rotating and rotating cylinders exposed to the inner pressure. The effect of the angular velocity is investigated on the radial distribution of various components. The mentioned structure can be considered as a sensor for measuring the angular velocity of the cylinder subjected to the pressure and temperature. The obtained results indicate that the electrical potential is proportional to the angular velocity.