Thermoelastic analysis of functionally graded annulus with arbitrary gradient

A thermoelastic problem of a circular annulus made of functionally graded materials with an arbitrary gradient is investigated. Different from previous works, our analysis neither requires a special form of the gradient of material properties nor demands partitioning the entire structure into a multilayered homogeneous structure. Instead, we propose a new method for solving the thermoelastic problem of a functionally graded circular annulus by transforming it to a Fredholm integral equation. The distribution of thermal stresses and radial displacement can be obtained by solving the resulting equation. Illustrative examples are given to show the effects of varying gradients on the thermal stresses and radial displacement for given temperature changes at the inner and outer surfaces. The results indicate that the thermal stresses can be relaxed for specified gradients, which is beneficial to design an inhomogeneous annulus to maintain structural integrity.     

内含弹性介质功能梯度压电球壳的径向振动调控

针对内含弹性介质的功能梯度压电球壳的径向振动问题,利用变量替换技术,求得了解析解。该解适用于材料参数沿厚度按幂律变化且密度的梯度指标可与其它物理量的梯度指标不同的功能梯度压电球壳,克服了以往分析中所有材料参数梯度指标均假设为相等的局限。数值结果表明,功能梯度压电球壳的径向振动特性可利用材料参数梯度指标和内部弹性介质的刚度进行有效调控。     

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.     

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.     

Transient thermal stresses in two-dimensional functionally graded thick hollow cylinder with finite length

In this paper transient thermal stresses in a thick hollow cylinder with finite length made of two-dimensional functionally graded material (2D-FGM) based on classical theory of thermoelasticity are considered. The volume fraction distribution of materials, geometry and thermal load are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure. Temperature, displacements and stress distributions through the cylinder at different times are investigated. Also the effects of variation of material distribution in two radial and axial directions on the thermal stress distribution and time responses are studied. The achieved results show that using 2D-FGM leads to a more flexible design so that time responses of structure, maximum amplitude of stresses and uniformity of stress distributions can be modified to a required manner by selecting suitable material distribution profiles in two directions.     

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.     

The axisymmetric partial slip contact problem of a graded coating

In this paper, the axisymmetric contact problem with the partial slip condition for a functionally graded coated half-space which is indented by a rigid spherical indenter is considered. The material properties are assumed to vary along the thickness of the coating. A series of linear functions of the thickness are used to model the functionally graded coating with the arbitrarily varying shear modulus. The contact problem is formulated in terms of a set of Cauchy singular integral equations by employment of Hankel integral transforms technique and transfer matrix method. By using the uncoupled solution and the coupled solution,the coupled equations are solved. The effect of the coating’s gradient on the normal and radial tractions in the whole contact region is presented. The results show that the contact tractions and the size of the stick zone can be altered by adjusting the gradient of the coating. This may have potential applications in the resistance of contact deformation and damage.     

Stress analysis for a coated fiber embedded in an infinite matrix subjected to body force

A micromechanics model of a functionally graded coated fiber which is embedded in an infinite matrix is presented. Various combinations of friction-less sliding and perfect bounding conditions at interfaces are considered. The fiber loading is assumed to be a distributed body force. The mechanical properties of the graded coating are assumed to vary smoothly and continuously with the change of volume concentrations of the constituting materials along the radius according to a power law distribution. Numerical results show that the hoop stress at the fiber boundary in the matrix can be reduced to becoming negligible if an optimum volume fraction is chosen for coating.     

AXISYMMETRIC BENDING OF TWO-DIRECTIONAL FUNCTIONALLY GRADED CIRCULAR AND ANNULAR PLATES

Assuming the material properties varying with an exponential law both in the thick- ness and radial directions,axisymmetric bending of two-directional functionally graded circular and annular plates is studied using the semi-analytical numerical method in this paper.The de- flections and stresses of the plates are presented.Numerical results show the well accuracy and convergence of the method.Compared with the finite element method,the semi-analytical nu- merical method is with great advantage in the computational efficiency.Moreover,study on ax- isymmetric bending of two-directional functionally graded annular plate shows that such plates have better performance than those made of isotropic homogeneous materials or one-directional functionally graded materials.Two-directional functionally graded material is a potential alter- native to the one-directional functionally graded material.And the integrated design of materials and structures can really be achieved in two-directional functionally graded materials.     

轴对称任意梯度圆筒的弹性动力学解

基于轴对称平面应变问题的运动方程及弹性梯度材料的应力和位移关系,通过将圆筒分层使材料性质离散为分段常数函数,同时在时域内应用有限差分格式,求得了材料性质沿径向梯度变化的圆筒弹性动力学解。本文解不仅适合任意梯度的弹性圆筒,而且容易满足多种形式的初始条件和边界条件。通过对材料性质沿径向为连续函数分布和分段函数分布的梯度圆筒数值分析,并与已有文献结果比较,得出本文解与已有文献的解吻合较好,验证了本文解的正确性和有效性。对材料性质为分段函数的三层组合圆筒分析发现,中间功能梯度层的指数分布因子对圆筒的径向位移和应力随时间变化都会产生显著影响。     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations

Three-dimensional free vibration analysis of functionally graded piezoelectric (FGPM) annular plates resting on Pasternak foundations with different boundary conditions is presented. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the influences of the Winkler and shearing layer elastic coefficients of the foundations on the non-dimensional natural frequencies of functionally graded piezoelectric annular plates. The analytical solution in the thickness direction can be acquired using the state-space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the convergency and accuracy of the present method. The influences of the material property graded index, circumferential wave number and thickness of the annular plate on the dynamic behavior are also investigated. Since three-dimensional free vibration analysis of FGPM annular plates on elastic foundations has not been implemented before, the new results can be used as benchmark solutions for future researches.     

Thermal stresses in functionally graded materials caused by a laser thermal shock

 Mathematical simulation of a thermal shock method for reliability testing of functionally graded material (FGM) is performed with the end to determine operating parameters of the testing device (power of a laser, laser beam radius, duration of heating) and to investigate the effect of the composition of FGM on a magnitude of thermal stresses in a coating. An analytical method for solution of the thermal elasticity problem is developed whereby the approach of a multilayer plate is used for determining temperature and thermal stresses distributions in a coating. We considered the limiting case of the obtained solution when the thickness of a layer is infinitesimally small and the number of layers tends to infinity. This procedure allowed us to obtain the thermal stresses distribution in a FGM coating. The results for the FGM coating composed of WC (tungsten carbide) ceramics and HS-steel are presented. It is showed that variation of the volume content of ceramics strongly affects thermal stresses in a coating and they decrease significantly in the case of the uniform spatial distribution of ceramics. Received on 21 November 2000 / Published online: 29 November 2001     

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 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.     

Fracture analysis of functionally graded coatings: antiplane deformation

A new model for approximate analysis of a functionally graded coating with arbitrary variation of properties has been developed. In this model, the coating is divided into several sublayers. In each sublayer the material properties vary in a linear manner along the thickness direction and the material properties are continuous at each subinterface. With this new model, the crack problem of a functionally graded coating bonded to a homogeneous half-plane under static antiplane shearing load is investigated. By using transfer matrix method and Fourier integral transform technique, the problem is reduced to the solution of a Cauchy singular integral equation. Stress intensity factors are calculated. Comparisons are made between the present model and other models. The results show some advantages of the present model.     

Three-dimensional free vibration analysis of four-parameter continuous grading fiber reinforced cylindrical panels resting on Pasternak foundations

This research investigates three-dimensional free vibration analysis of four-parameter continuous grading fiber reinforced (CGFR) cylindrical panels resting on Pasternak foundations by using generalized power-law distribution. The functionally graded orthotropic panel is simply supported at the edges, and it is assumed to have an arbitrary variation of matrix volume fraction in the radial direction. A four-parameter power-law distribution presented in literature is proposed. Symmetric and asymmetric volume fraction profiles are presented. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are solved by generalized differential quadrature method, and natural frequency is obtained. The fast rate of convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for functionally graded isotropic shells with/without elastic foundations. The effect of the elastic foundation stiffness parameters and various geometrical parameters on the vibration behavior of the CGFR cylindrical panels is investigated. This work mainly contributes to illustrate the influence of the four parameters of power-law distributions on the vibration behavior of functionally graded orthotropic cylindrical panels resting on elastic foundation. This paper is also supposed to present useful results for continuous grading of matrix volume fraction in the thickness direction of a cylindrical panel on elastic foundation and comparison with similar discrete laminated composite cylindrical panel.     

Analysis of functionally graded rotating disks with variable thickness

Elastic solutions for axisymmetric rotating disks made of functionally graded material with variable thickness are presented. The material properties and disk thickness profile are assumed to be represented by two power-law distributions. In the case of hollow disk, based on the form of the power-law distribution for the mechanical properties of the constituent components and the thickness profile function, both analytical and semi-analytical solutions are given under free–free and fixed-free boundary conditions. For the solid disk, only semi-analytical solution is presented. The effects of the material grading index and the geometry of the disk on the stresses and displacements are investigated. It is found that a functionally graded rotating disk with parabolic or hyperbolic convergent thickness profile has smaller stresses and displacements compared with that of uniform thickness. It is seen that the maximum radial stress for the solid functionally graded disk with parabolic thickness profile is not at the centre like uniform thickness disk. Results of this paper suggest that a rotating functionally graded disk with parabolic concave or hyperbolic convergent thickness profile can be more efficient than the one with uniform thickness.     

均布荷载作用下功能梯度简支梁弯曲的解析解

利用应力函数半逆解法,研究了均布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲的解析解,给出了各向应力应变与位移的解析显式表达式.首先根据平面应力状态的基本方程,得出了功能梯度梁的应力函数应满足的偏微分方程,并根据应力边界条件得出了各应力分布的表达式;进而根据功能梯度材料的本构方程和位移边界条件,得出了应变和位移的分布.最后,通过将本文的解退化到均质各向同性梁并与经典弹性解比较,证明了本文理论的正确性,并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布,分析了上下表层材料的弹性模量比λ与组分材料体积分数指数n对应力和位移分布的影响.     

Fretting contact of two elastic solids with graded coatings under torsion

In this paper, the fretting contact problem for two elastic solids with graded coatings is investigated. We assume a conventional axisymmetric Hertzian contact takes place between two elastic solids under the action of the normal pressure. The application of the torque produces an annulus of slip. It is assumed that the surface shear traction within the contact area is limited by Coulomb’s friction law and the torsion angel was produced within the central adhesion zone as a rigid body. The linear multi-layer model is used to model the functionally graded coating with arbitrarily varying shear modulus. This model divides the coating into a series of sub-layers with the elastic modulus varying linearly in each sub-layer and continuous on the sub-interfaces. By using the transfer matrix method and Hankel integral transform technique, this problem is formulated as the solution of the Cauchy singular integral equations. The contact tractions are calculated by solving the equations numerically. The results show that the appropriate gradual variation of the shear modulus can significantly alter the contact tractions. Therefore, graded coatings may have potential applications in improving the resistance to fretting contact damage at the contact surfaces.